He probably should have just set a deadline: Said that the Senate has one month to "advise and consent" and if they chose not to then Garland would be seated. What could Mitch do? If Garland showed up at the Supreme Court, put on a robe, and started to hear cases with the acceptance of the other justices, that would have been a done deal. It's not like the Senate has any law enforcement officers they can send to the Supreme Court. All he could have done is sued, and that would have taken a very long time and would have to go through several layers of federal courts, only to maybe end up at the Supreme Court which had already accepted Garland as a centrist Justice. But I think the various federal judges would have seen it as the Legislative Branch attacking their Branch.
Anyway, it would have been fun to see the Supreme Court ruling on itself, in effect, sort of like Marbury v Madison.