I unsuccessfully attempted to separate the differential equation problem from the algebraic problem.

I had a different idea that you might see if you could use. Differentiate both sides of your Eq111. That gives you the needed differential equation for H[z]. Since that adds another differential equation, you may or may not need another initial condition to uniquely reconstruct H[z].

When I took a moment and tried to do this I ended up with multiple warnings about 1/0. I didn't explore whether it might be possible to avoid those by getting rid of denominators or if this was because of something else. But if you can track down the cause of those complaints and resolve them then this might get you where you need to go. Or at least you might learn a little more about your system.

With a large magnifying glass and carefully squinting, notice the font of the \[Psi]''[10] inside your NDSolve is microscopically different from other instances, like the \[Psi]''[z] that begins your Eq222. This means you have been "desktop published" and while your equations might look good they are unfortunately meaningless to Mathematica.

To try to fix this, if you carefully scrape the \[Psi]''[z] that begins your Eq222, which appears to be OK, and paste it over the top of the \[Psi]''[10] inside your NDSolve and then carefully manually replace that z with 10 then the error message you see goes away.

Unfortunately it is then replaced with two other messages. Those are less related to desktop publishing, more directly related to mathematics and with a little math background you might be able to better interpret those messages. The first message is cautioning you that you have mixed equations that have no derivatives with equations that do have derivatives and Mathematica will try to deal with this. The second message is telling you that it thinks you have given it a boundary problem and with mixed equations it can only deal with initial value problems when given mixed equations. That may or may not be a misunderstanding by Mathematica or by you.

Can you perhaps separate the problem of solving the differential equations from solving the algebraic equations? That might help avoid some of the difficulty.