I would expect that the standard of evidence here would be higher than an estranged wife's bitter social media posts.
And apart from some kind of a criminal investigation, I don't see how the NHL could compel the bookies to give up their secrets, assuming they could even find these bookies.
On the other hand, if Kane was stupid enough to bet using legal means then he could be in a ton of trouble. But that isn't the allegation here.
As for the ex, I assume that she will wisen up to the fact that having Kane banned from the NHL is not in her best interests. His further employment in the NHL is her only hope for financial stability in the future for her and the kids.