This is a passage from NHK's 歴史秘話ヒストリア about Oda Nobunaga performing Noh before the Battle of Okehazama:
From the dictionary the only usage of をつける with a person seems to be:
But that doesn't translate go "Nobunaga put the efforts as he was performing in front of a master", imho.
And the phrase meaning seems more like Nobunaga was achieving the performance level of a master. But it's not に着ける.
It also doesn't seem to be a set phrase for performing arts - Google returns only the old transcript of the very TV programme.
How to interpret the phrase 師匠をつけるほど here?