In a LED the electrons from the n-type combine with the p-type which creates a photon. Are the electrons 'lost' or gone when this process takes place?

By 'combined' with the holes in the valence band (not shell), you mean the recombination process. To answer your question, you must understand the concept of 'hole'. You probably already know 'hole' is not a separate particle rather it is a lack of electron. In overly simplified words, you can think a positively charged ion has a hole in it as it requires a electron to neutralize itself. If a electron is near by, the ion attracts it, captures it and becomes neutral and in the process emits a photon. In terms of 'hole' we say, electron recombines with the 'hole' and emits a photon. But note that the captured or recombined electron is not vanished. It sits in the electron shell of the atom. Thus it does not vanish, rather goes to a bound state within the atom.

From the above explanation, I guess the question of something 'running out' of electron is no longer a problem. The electron does not vanish. It just stays within the atom. The emitted photon is just the excess energy carried by the electron. Due to quantum nature of the electron's orbits, the emitted photons have discrete frequencies. If you have further confusion, let me know in the comments, I will be glad to answer it.