All irreducible representations of ${\rm GL}(2, {\mathcal O})$ have been constructed by Alexender Statinski :
Stasinski, Alexander (2009) The smooth representations of ${\rm GL}(2, {\mathcal O})$.
Reference in Math. Arxiv : http://arxiv.org/abs/0807.4684
But he does not follow the procedure that you propose at all. He uses much more generalizable tools : Clifford theory and an adapted version of Kirillov's orbit method. These ideas are now classical in type theory for $p$-adic reductive groups (see e.g. Howe and Kutzko's works).
Some people are now working on the more general question of constructing the representations of ${\rm GL}(n,{\mathcal O})$ (A. Stasinski, A.--M. Aubert, F. Courtès, and others). This problem is known to be "wilde"wild"
