The solution that came to mind was typing (with a text-to-speech or text-to-braille solution for the blind person).

That seems to be the obvious solution. The part that makes me intrigued, however is how the increased overhead of verbal communication would encourage a heavily intuitive physical language to emerge. Even more fascinating would be if the participants started their interacting as children. I would expect a full physically mediated grammar to evolve.

If the deaf person could read lips and speak understandable English (and some can), they could just talk.

I distinctly remember typing 'deaf and dumb'. I must have edited that out while making the phrasing fit.