I need to write a letter to Shakespeare explaining why I appreciate him for the methods used by Petruchio to tame Katharina in The Taming of the...

This is a good assignment that is being used to test your
knowledge of the play but it is also after your own opinion and thoughts and it is
testing your ability to express those in a letter form. So, the best place to start is
for you to think about what you thought was funny or particularly effective about the
methods that Petruchio adopted to "tame" his wife. When you were reading or watching the
play, what amused you? I would say that you probably need about three different examples
from the play, each of which that can form a middle paragraph of your essay. For each
paragraph you need to offer a brief summary of what Petruchio did and then say briefly
why you think it was effective. Consider the following
example.

I loved the way that, in Act IV scene 1, Grumio
relates to Curtis how when Katharina's horse fell and she ended up in mud, Petruchio did
nothing to help her but only beat Grumio so that Katharina had to wade through the mud
to pull him off Grumio. That was an excellent way of humbling Katharina and also to stop
her thinking so much about her own needs and desires. Beating Grumio forced her to think
of others for a change.

This is just a quick example, but I
hope you get the idea. Good luck!