Every kimchi class ends the same way. Someone holds up their jar and asks what happens next. The honest answer is that the work is mostly done and the jar takes over from here.
Leave it on the counter for two to three days. In a warm Valencia kitchen it can be ready in less. You are looking for bubbles around the edge and a smell that has turned sharp rather than raw. Press the vegetables down under the brine each day, because anything sitting above the liquid will spoil long before the rest.
Once it tastes sour enough for you, move it to the fridge. Fermentation does not stop there, it just slows to a crawl. A jar that tastes bright in July will taste considerably deeper by September, and both are correct. If it gets away from you, that older, funkier kimchi is exactly what you want for a fried rice or a stew.
The only real failure is mould on the surface, which is always a sign that something was sitting out of the brine. Everything else is just the jar doing its job.