The maximum bonus you can achieve for an 8pt base-magnitude enchantment (Fortify School, Alchemy, Smithing, etc) in the base game, reached after a few rounds of the "crafting loop" of using Enchanting to make +Alchemy gear and then using that gear to make +Enchanting potions, is 35%.

Dragonborn changed that with multiple bonuses, most of which require even more grind as part of the cycle with two trips to Apocrypha for each pass through the loop. The "simple" bonus is Azhidal's Genius, which at +10 POINTS of Enchanting skill is almost (but not quite) worthless, but does push the endpoint to 36% per item. The much larger and more important bonus though is actually somewhat hidden: although it's obvious that Seeker Of Shadows should give a 10% bonus to Alchemy, it's a lot less obvious that Seeker Of Sorcery will do the same for Enchanting. It's not shown at the time you choose it, only after you HAVE chosen it and examine the description of the power.
Regardless, that bonus allows you to break through the normal "base" limit of +36% per item, not by just 1 more point like Azhidal, but to a staggering +40%.

That in turn means that the Alchemy and Smithing bonuses on the crown should be a very impressive +160%, even without going through the crafting loop yet again (after also switching back to Seeker Of Shadows before making the potions). The next pass through the loop results in potions of +43% Enchanting, and +41% Alchemy per piece, but before I continue with that grind...

Things get tricky here, because since the player is guaranteed to have ONE of the "Secret Of X" bonuses once they've done that book, the crown itself needs to NOT include whichever bonus that is to avoid stacking them. If it factors that in blindly to give +176% to potions, then any Alchemy done while also having Shadows active would get a doubled bonus. The same goes for the base Smithing bonus and Seeker Of Might.

Even so, with the original version of the crown equipped, with Sorcery but no potions, enchanting an item with +Alchemy currently "only" results in a +38% bonus, which is less than what can be made without using the crown at all and just grinding through the crafting loop WITH the book bonuses. So regardless of anything else, the +Ench bonus on the crown should actually be even higher than it already is. That's quite noteworthy, because since the "Fortify School" settings are the same magnitude as Alchemy it means you can get 80% cost reduction from just TWO equipment slots!
While that's still some way from true "freecasting" it's close enough that for anything except Destruction (where you need to spam spells constantly) it pretty much removes any need to invest in Magicka at all (if for some reason you weren't already at that point even with the old one). At 100 Restoration, the cost of Grand Healing drops from 151 to *27*, without taking any perks. Even Bane Of The Undead, which at 586 Magicka is insanely expensive, only costs 105 with just two enchantment slots taken (and obviously, perks or Sorcery would reduce that even further).

The next round brings potions up to +44% Enchanting, but that isn't enough to improve the items past +41%, so we're finally done with the loop.

The mess of "stuff" that DB added is a major headache. There are really only two options:

1) Nerf the crown's bonuses so that you have to use the appropriate "Seeker Of" bonus too. That means traveling to Solstheim basically every time you want to craft anything, and almost certainly going through at least 2 trips to Apocrypha every time and potentially even more. Clearly, this option sucks hard.  :(

2) Try to cope with the Seeker bonuses via additional Conditions on the crown (which is how the 25%/40% choice via the pickaxe is handled). This makes things a LOT more complicated, since each component of the enchantment would have to be duplicated with a has/hasn't adjustment for each of those bonuses. That's manageable, but gets even less so if I want to keep the nerfed Enchanting option available too (which I do, since that's the one I use myself).
This is undoubtedly going to confuse some players, who will wonder why Of Might etc "isn't working" for crafting, but that's life.

To make #2 work, the "base" bonuses for Alchemy and Smithing need to be increased to +164%, as the minimum for them, and then they also need versions at a further +10% (which in this case is a "real" 110% net result, i.e. +180% total, rather than just adding another 10 points for +174%) if the player does NOT have the corresponding Secret Of Whatever active.

Unfortunately, we've reached a point where there are so many pieces in play, and one of them is so broken, that we can't actually get to the right results the right way.  :(
With 4 pieces of +41% Alchemy gear (=164) and Shadows (x1.1, =180) I made +44% Enchanting potions. With a flat +180% bonus from the crown, the potions are now only +42%. That's because the Black Book adjustments are "Skill X is 'more effective'", which isn't the same thing as either +10% to OR 110% of the skill value. That sort of manipulation of the "secret" internal values is ONLY available to Perks, so there's no way to achieve the same result the same way with an Enchantment. I can think of 3 options here:
1) Abandon trying to cope with the Black Book effects, dooming players to pointless trips to and from Solstheim and constantly re-reading Black Books.
2) Create one or more new hidden perks to match the same math.
3) Add random numbers to the +180% until even the most "sensitive" potions match up.

#3 is a lot harder than it sounds: Fortify Smithing is "sensitive", at around +160%, but there are things like Stamina Regen effects that are currently at 198%. The multipliers are so large, and the math so bizarre, that it takes a lot of slow guesswork to even get close, and since the underlying equations are different there will undoubtedly be cases in some of the multi-effect potions where rounding happens in different places and the results don't quite match up.

That really only leaves #1 or #2, and since I've basically finished my current playthrough I'm reluctantly leaning towards #1 for now. There's definitely far too much screwing around involved in it - just look at all the extra steps involved:
* Travel to Solstheim
* Read the book, choose Shadow
Make Smithing potions
* Read the book, choose Might
Smith your gear
* Read the book, choose Sorcery
Enchant your gear
* Read the book YET AGAIN, choose whatever your actual bonus is
* Travel back to Skyrim

Everything starred is gratuitous petty grind, and not only is it 2/3 of the whole cycle, but each of the stages it adds is slow. Since it's so bloated, it's also very easy to make a mistake at some point.
But in my current playthrough I just ignored all the Dragonborn crafting mess until I'd maxed all 3 crafting skills, when I made a single trip there to create one set of endgame gear each for me and my follower, and I didn't feel like I was really missing out on anything. By the time you're doing 500+ damage with a dagger, bumping that to 600+ instead barely matters even on Master.

I think it IS "correct" to upgrade the "base" bonuses for Alchemy and Smithing to 164%, because even though you have to have completed Azhidal and Dukaan to be able to make +40%-plus gear, once you've done that you HAVE that gear, and the gear itself doesn't get any weaker just because you chose a different bonus later on. Fundamentally, the same holds true for Enchanting: you can make a giant stack of +44% potions and use those regardless of any other factor, so again, the crown should echo that. (Especially for that case, since that's the one with the pettiest timer and the clumsiest UI).

I'm comfortable with just making those changes for now, and leaving it at that. The whole Dragonborn mess is exactly that, and getting caught up in trying to make it so that the crown is the ONLY thing you need, as it originally was, can't be done any more without a lot more complexity than I really want it to have. Not impossible, and really not even all that difficult, but clumsy and ugly.

Update
To reach the "full" Enchanting bonus WITHOUT have to screw around going to Solstheim and making multiple trips into and out of Apocrypha, you need Fortify Enchanting potions of EXACTLY +20% on top of the Crown's bonus. By sheer luck, it turns out that requires Fortify Alchemy gear totalling +31%, which happens to be exactly what you get if you use the Crown (without the Pickaxe) once both skills are maxed. So while that does require a second item, it's one that's very easy to make, and once you've made a stack of +20% potions you can just stash them at home and use them whenever. Certainly a lot quicker than going through all that travel every time.  :)