There are a few common issues that will cause this to happen.
One of them is that the solder joint on the headphone jack where all the black wires meet is sometimes hard to solder, and if one wire is loose in that joint, then you'll get abnormalities like you describe.
Another issue is that the heater supply for the tubes (the twisted pair of wires that goes to A4/5/9) must heat the tube up in order for it to conduct, and currently your tube isn't conducting. You can look at the 12AU7 while the amp is running to see if it's glowing as a check.
What I wouldn't do is obsess over the LED's themselves, your issue is almost certainly elsewhere in the circuit.
-PB