Ghost is very similar to Cake in it's writing and commenting interface, it's a joy to use, but it's not really suitable to run a multi-poster and many comments site like Cake at all.
NodeBB is based on the same kind of event driven node.js architecture as Ghost, which makes is really fast compared to most PHP based forum software, and it would be an option that would need a lot less work.
Aside from that there is forum software. Xenforo, of course, which is a more "traditional" forum platform I guess, but also something like Discourse or Flarum. We are in the middle of implementing a Flarum based forum for a client and I'm really liking it.
Personally, if it were my project, I would seriously be looking at the options for reducing the maintanence and dev costs of Cake as it runs now but staying with the custom codebase that it has. I'd open-source the code and try to attract people to participate in developing new features, reduce reliance on any expensive external dependencies and replace things that can be replaced with good well maintained opensource libraries.
I love the look and feel and function of Cake as is, and that would be the best way to preserve and build on those things.
If that's not an option then any of the things I've listed above would be a good base for building a new version. I'd probably shy away from Discourse because it's heavy in resource use and it just looks like Discourse which is everywhere already. NodeBB, Flarum or Xenforo would be fine though.