Thanks for the info AbhishekS!
My only concern is this: I am not sure if just the outer screen is broken, or the digitizer as well.
For now I bought only the outer screen, which is cheap (10 pounds).
My problem is this: If the digitizer is broken, then I need to replace that part as well, but it is expensive! It is so expensive that it is probably more worth it to buy a Nexus 4 or some other better android device at this point.
So I am facing two options: Either try to replace just the screen or.. buy another phone. Replacing just the screen seems way harder that replacing the screen+digitizer combo and I am kind of afraid to do it... :/