What is kimchi "brine"? So I made kimchi for the first time a couple of days ago, and most of the recipes that I looked at were fairly similar. In this regard, all of them mentioned something that I'm not 100% sure on what it means.

So I followed the standard steps: I rubbed the cabbage leaves with salt until they started to wilt, soaked it in salted water for a few hours, drained and rinsed the cabbage, squeezed out most of the water from the leaves, and then combined them with the rest of the ingredients and chilli paste before putting it all into a fermentation jar.

However, every recipe that I saw said that once the mixture was in the jar, I needed to push it down until the brine was covering the vegetables. I pushed it down but, other than getting rid of the air-pockets, it didn't seem that there was any "brine" that was covering the vegetables.

Worrying that I had squeezed too much liquid out of the cabbage leaves and this was the "brine" that it was referring to, I added a little more water back in until it was just about covering everything.

The recipe also stated that I should check the kimchi daily to make sure that the vegetables were still beneath the brine. However when I went back to check, there was a large amount of liquid above the vegetables. Now I'm worried that adding more water to the mixture was a huge mistake.

Did I mess up? And if so, can I just drain off the excess liquid and fix the issue? Or is it fine that there is a lot of water covering the vegetables?
This is not simply about water or liquid covering the veg, but brine (salted liquid) covering the veg.  The brine is necessary for fermentation and safety. I would worry that your rinse and soak steps washed away too much salt. There is no way to know for sure from your description.  When I make kimchi, I salt the cabbage, and leave in a colander overnight for excess moisture to drain.  Then without rinsing, I proceed with the rest of the ingredients. I pack into a container.  There is typically enough brine to keep the ingredients in liquid for the fermentation to proceed safely.  If you have kept yours refrigerated, eat it. If it has been at room temp, with low to no salt, I would be concerned.