Decidability of Unary Languages / One-to-One Mapping

Question

I'm trying to prove that there exists an undecidable subset of {1}* by showing a one-to-one correspondence between it and {0, 1}* (which would imply a one-to-one correspondence between their power sets), but I'm struggling with how to do the one-to-one mapping. Isn't it just surjective? That is, there's one unary representation of potentially many binary strings (e.g., 1 = 01 = 0000000000001).

What am I misunderstanding here? Or am I just taking the wrong overall strategy?

(This isn't homework; I'm reviewing for a midterm, and it's a little concerning I'm getting tripped up here)