I’ve actually found a thread where this exact phrase is discussed on these forums, so I’ll linked it directly:
As for these words,
かぎって is just the てform of かぎる、which means “to be restricted, to be limited”, so この夜にかぎって would mean “only for tonight”, or something.
はせず: ok so the は here is the particle wa and it contrasts the 眠ろうとせずに with the next sentence. せず here is the ず conjugation of する. せずに means the same as しないで in modern Japanese. 眠ろうとする is the “volitional + to suru” conjugation which means “to try to …” so in the end 眠ろうとはせずに would mean something like “without trying to sleep, instead(this is from the contrasting は)…”
いつまでも means “forever, for good, eternally, as long as one likes, indefinitely, no matter what”
So, as a whole, “眠ろうと はせず に、 いつまでも 、ランプの下にすわって起きていたのでした。” would mean
"Without trying to sleep, instead (he) always sat awake at the top of the ramp(lamp maybe?).