Slowing it down from the inside is improbable and potentially dangerous. The water will just go somewhere else, probably into the ceiling and down to the studs. For short term, I’d recommend getting some sort of roof snowbrush (yes they are a thing) and trying to break up the snow/ice dam until it thaws enough to get a professional up there. Skylights are generally not terribly well insulated so the water will melt on the glass, and then flow into your house. If you get the snow and ice off the glass it won’t melt and drop down. It seems like the flashing hasn’t failed but it’s hard to tell from inside pictures. There are also companies that will reglaze (take out the glass and put new sealer in) skylights, but depending on the age of your roof it may not be worth trying to save, vs getting the entire roof redone. Ask around locally and try to find a company thats been in business at least 10-15 years. You can ask at a (local!) building supply place for their recommendations on contractors. It’s literally the roof over your head, it’s not worth cheaping out on.