For the parabolically induced representation, I suggest to look at Casselman "Restriction of $GL(2, F)$ to to $GL(2,o)$"-paper.

For the Steinberg representations and the super cuspidal representation, I suggest to look at Bushnell-Henniart "Local Langlands conjectures for GL(2)". For the Steinberg, your ideal will be the maximal ideal $p$. For the supercuspidal stuff, you should try to understand the definition of a stratum.

The translation from strata stratum for super-cuspidals to what you are asking about is essentially an a time-consuming exercisewhich . It suggest to rather argue in terms of isotypic subspaces than in terms of invariant vectorswith strata directly.

Also this article by Ralf Schmidt seems relevant: http://www.math.ou.edu/~rschmidt/papers/gl2.pdf.

Silberger has also classified representation of GL(2) in "Representations of PGL(2) over the $p$-adics" (LNM).

I am not sure if the titles of the references are all correct.

For the parabolically induced representation, I suggest to look at Casselman "Restriction of $GL(2, F)$ to to $GL(2,o)$"-paper.

For the Steinberg representations and the super cuspidal representation, I suggest to look at Bushnell-Henniart "Local Langlands conjectures for GL(2)".

The translation from strata for super-cuspidals to what you are asking about is essentially an exercise which suggest to rather argue in terms of isotypic subspaces than in terms of invariant vectors.

Also this article by Ralf Schmidt seems relevant: http://www.math.ou.edu/~rschmidt/papers/gl2.pdf.