Splittet Value Function and Hamilton-Jacobi-Bellman equation
I don't think I've seen cases with a kink in continuous time that are solved numerically. Aguiar &amp; Amador (2014) have a kink in their HJB, but they solve their HJB analytically. 

Validation

I don't know to what extent Moll et al's algorithm is going to get that right. You will most likely get weird behavior around the kink, but whether it's the real functional form is to be doubted, and you'll have to argue ex-post that "it looks okay alright?". 

If you want to get a better grip into whether the kink looks alright, make it more extreme. Without going through your equations, it seems that the first part is convex and the second part is linear. Increase the curvature for the first part so you can "see" more easily whether it still looks correct around the kink.

Also, you definitively should look at the numerical counterparts to the first-order and second-order derivative, the first-order and second-order difference of the approximation divided by the step size. Do they have the correct signs around the kink?

One thing I'd emphasize is having many grid points around the kink. However, since the algorithm gets tougher when you have a nonlinear grid (because the grid step size is now not a constant, but a vector), it might just not be worth it and you could rather jack up overall grid points and keep it linear.

Speed

Finally, you really should look into having sparse matrices for the A and B matrices. 10 iterations per minute is really slow. It takes you perhaps 30 minutes to understand and implement (if you've never worked with sparse systems before), but already after the second time you run the code you've made up that time. 

If the problem persists, have a look at profilers for your coding language. For Python, this is a nice solution. Even if you have everything sparse, perhaps you forgot some matrix that you don't even need in the iteration loop. That was the case for me, and with large grid sizes that single reinitialization took up a lot of time. Profiler will tell you what's what.