The adapter is becoming warm, but the speaker
The last point is the most important -- the speaker light is not glowing. That means it's not getting power, and nothing else is going to work.
If you have access to a multimeter (voltmeter) you can test the adaptor to see if it's producing the power that is on the label. That would help you figure out if the problem is the adaptor or inside the speaker.
If you bought it retail, I'd suggest taking it back. I can't say for sure where the defect is, but "I plug it in and it doesn't turn on" is a sure-fire indicaor that it's defective.