Re: My boat livewell fills and then overflows into the boat, but I don't know why.
You lucked out there with easy access, Nice!
As others have stated, make sure to seal up the thru-hull with 3M 5200, put a healthy line on the fitting and nut, snug it down and then clean off the excess on the exterior of the hull with paint thinner, WD-40, etc. When you go to attach the hose, do the same, jam some sealant (not much) into the hose, smear it around with your pinkie, and then attach the hose. (don't use silicone)
Something else, since the drain thru-hull broke, you might want to look at the livewell pump and the same time and see if it looks aged, deteriorating. If it's in question you might as well replace it too, Walmart has them for $30.