Shows

This might be a good start: https://www.londontheatre.co.uk

Theatremonkey is also useful. I like the information about the seating in each theatre (and sometimes how it affects sightlines for specific productions).

Yes good places to check. But, buy your tickets directly from the theater. I did a mock booking for Book of Mormon and the same 2 seats in the stalls (front orchestra) were over 100 GBP less through Delfont Macintosh theaters who own the Prince of Wales theater as well as several others.

I think The Woman in Black is still showing and is one of the cheaper matinee shows. Excellent. Ignore the movie, it may put you off.

If your research results in a burning desire to see a specific show but it seems to be sold out by the time you are ready to buy a ticket, you might opt to wait and keep checking. I was able to pick up seats to a couple of sold-out performances after I arrived in London by checking for last-minute availability on the internet each evening. I wouldn't count on that working for "Hamilton", but even that would be worth trying.

I will always advocate for Wicked if you haven't seen it. I had already seen it before I went to London but jumped at the chance to see it again. My friends went to Kinky Boots and said it was amazing. So that one could be an option.

If you don't have your heart set on a specific show, then visit the TKTS booth. You can get an idea of what they're likely to have on sale from visiting their website, but tickets have to be bought in person.

https://officiallondontheatre.com/tkts/

The Globe would be an obvious choice. The Bridge theater also.
Personally, I would be looking for something classical. a lot of well know British actors will appear in very small productions in London

Kinky Boots, Wicked, Hamilton
Folks, you are going to London, not Vegas.
Sort of going to Paris and asking where the McDonald's is
Take the opportunity to see a level of theater that you have not had the chance to experience.