Is there a website that simplifies the process of computing the private key from a weak signature?
I doubt it. There is a website that explains the math behind it, but it isn't a step by step guide or even an automatic JavaScript program.
Note that there are very few broken signatures, maybe once every few months. And they are usually exploited quickly (which is only possible if the addresses are reused). And some people have bots running that can do this immediately.