There’s really nothing you can do except cut away all the truly dead stuff, replant it and water it. Then cross your fingers. If it survives you can worry about pruning it back into shape in due course. You might try a few cuttings as insurance.

But .... do you know what it is? Is it honestly something worth keeping?

Sort of looks like Trachelospermum. That will survive almost anything, although seems easier to just go buy a new one if you really want it (very strongly scented white flowers, lovely if you are not right on top of it, cloying if so).

It is probably not worth keeping, but at this point, it is a hobby just to observe how a plant like this comes back. It does have a very old and twisted trunk, and that seems worth keeping. It will probably end up as a container plant.

@Sara_Malone Trachelospermum is Star Jasmine? I wonder why this plant never grew very large even back when it was getting lots of water. Maybe the clay soil? What would be an ideal container soil for this? I read it wants fast-draining soil, but I also read it likes moisture (to thrive anyway). If you were using a 5-1-1 mix, you might want to use two parts of peat instead of one?

In the short-term, should it be in a different soil just to encourage roots to start growing?

I wouldn't try to grow this in a container as it's very vigorous. I am also unfamiliar with container potting soil other than cactus soil, which is what I use for conifers. My ID might be wrong, too, keep in mind. .