Radiator half-cold: Venting air stops after 20 seconds, no water comes out. Help?

I have an older radiator that is warm at the bottom/side near the pipe, but cold everywhere else.

I’ve tried the standard fix of venting the air using a radiator key on the opposite side, but I’m hitting a wall:

• When I open the valve, I hear a hissing sound for about 20–30 seconds, then it just… stops.
• No water ever comes out, even if I leave it open.
• I can hear a noisy "rushing water" sound inside the radiator while the valve is open.
• I tried the trick of turning all radiators in the house to max for 30 minutes before venting, but it didn't change anything.

Any advice on what to check next would be much appreciated!