It is not possible for the contract to be "I would" because the next verb is 'had'. Modals are always followed by base form - I knew I would have to do it. But it also doesn't make sense for the contraction to be 'had'. This would form the Past Perfect (had + past participle) which is used to talk about the past before the past. This is not established in this sentence. In short, both answers are wrong.

I knew (that) I HAD had to do that. Similarly: You knew (that) I had had to do that. I had had to do that - and you knew that! They knew (that) she had had to do that. She had had to do that! And they knew she had had to! So, I knew (in the past) that I had had to do something (further back in the past). They knew (in the past) that she had had to do something. She had had no choice. And they knew that!

Hii Avi, It's "i would". If you want to improve your fluency and speaking ability. Feel free to message me. Cheers :)

I'd means I had

I knew I would had to do that. This is obviously in speech or thought. It should correctly mean: "I knew I would have had to do that. " The person is omitting the have. I hope that helps. Kat the teacher