I wouldn't count on the broken-off vine rooting, but I'll bet the trunk is already throwing out new branches.

The main will probably grow laterals.
The broken part is a history, unless it had previously grown roots. Then you could cover them, water them, fertilize them ..

I feel for you. Today I was caging the last of the peppers and broke the top (flowering) of a jalapeno plant off just b/c I had 1 little branch under the bottom ring when I pushed down, though I was supporting the top of the plant to guide it through the ring. Should have done them when they were smaller, but some of the cages were in an old trailer I couldn't get open, needed DH to open the door.