Is 'soon' correctly used?

No.

You could say "We sat down to watch television but soon the lights went off", or "The television started to flicker, and soon the lights went off"; "Soon" is correct in both these sentences, because you have stated an initial event soon after which the lights went off. But "We were glued to the television" is a continuing state, so "soon" has nothing to refer to.

Another thing wrong with this passage is that "glued to the television" means "fascinated, completely absorbed in the programme"; but if the television starts to flicker, the effect on normal people is to disturb people and distract them from the programme. It seems very unlikely that you would still be glued to it when finally the lights went out!

You might recast this as:

My family and I were watching television when the lights and the television started to flicker. Soon the lights went off altogether - there was a blackout.

You could start with "My family and I were glued to the television" instead, but only if you particularly want to stress how absorbed you all were in the programme.

Thanks, Syntinen

The function of the word "soon" is used to give a sense of time in the list of events.

The people were watching TV; there was a flicker; then soon (afterwards) the lights went off.