To put it simple, the 5 years is for your security pass (DfT regulations). Any gaps (over two weeks) and you won't get a pass. If Jet2 know this up front then they won't waste time with you.
If I were you I'd get a pen and paper and start writing down all the details of every company, school, college, Uni or wherever you've been in the last 5 years. Exact dates required aswell. Jet2 have to write to each place individually, so if there's a few then it may take some time.
They're not looking for character references, at this stage, just proof that you were there for the dates given. Do all this and it will speed the process up and you'll get a pass nice and quick.
Good luck.