This is done in :

Casselman, William The restriction of a representation of ${\rm GL}_2 (k)$
to ${\rm GL}_{2}({\mathfrak o})$. Math. Ann. 206 (1973), 311–318.

Also see :

Silberger, A.: $PGL_2$ over the p-adics. Lecture Notes in Mathematics 166, Berlin-
Heidelberg-New York: Springer 1970

Very roughly speaking, the idea is the following. When you restrict an irreducible smooth representation of ${\rm GL}(2)$ to ${\rm GL}(2,{\mathfrak o})$ you get two types of constituents. The first ones (infinitely many) are uninteresting : they appear in the restriction of many other representations. These representations are described by Casselman. The second ones are more interesting : if an ireeducible representation contains such constituents then it must belong to a single component of the Bernstein decomposition of the category. They are called typical. Actually the situation is a bit more complicate. I've made it simpler. For more detail, you may read the paper by Henniart :

Breuil, Christophe; Mézard, Ariane Multiplicités modulaires et représentations de ${\rm GL}_2({\bf Z}_p)$ et de ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ en $l=p$. (French) [Modular multiplicities and representations of ${\rm GL}_2({\bf Z}_p)$ and ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ at $l=p$] With an appendix by Guy Henniart. Duke Math. J. 115 (2002), no. 2, 205–310.