So obviously we're not doing so great in puck possession at this point in time, but here's a graph of our 10-game rolling average of Fenwick %, if you need evidence:
This got me wondering: do Cup-winning teams go through similar slumps? To answer that question,
here are the same graphs for each of those teams since 2007-08. If you don't want to look at all of those graphs, here's a brief summary: only two of those teams went through similar slumps that put them below 50%: the Bruins and the Penguins.
2010-11 Bruins
2008-09 Penguins
Boston was truly awful until mid-November, and then had a slump similar to ours in late February through early March. They were kept afloat by Vezina winner Tim Thomas. In the playoffs, Thomas was unbeatable, and they got near-PPG efforts from Bergeron, Krejci, Marchand, and Horton.
The Penguins were terrible for most of the year, reaching their nadir in February, culminating with the firing of Therrien (see: 2015-16 Montreal Canadiens). But as soon as they hired Bylsma, they started improving, reaching 50% Fenwick in March and becoming a dominant team in April. In the playoffs, Fleury wasn't very good, but Malkin and Crosby put up insane numbers.
Let's compare this to our situation. Unlike Boston, our goalies aren't having Vezina-worthy seasons, and I doubt Allen or Elliott will set goaltending records in the playoffs. Other than that, though, their situation is fairly similar to ours. However, I think we can learn more from the Penguins. They (and the 2011-12 Kings) show that under-performing teams can reap immediate benefits from replacing the coach mid-season. The Kings are especially convincing, with their numerous playoff failures prior to their Cup-winning season.
TL;DR: Fire Hitch ASAP, and maybe we can do some damage in the playoffs.