Seek legal advice on this one. I worked for an holiday charter airline for many years. Believe me it is more than likely their legal people will chuck your correspondence in the bin, unless it has obviously not been concocted, i.e it looks like it could be serious and has an important letterhead, and was probably signed for on reciept. The CEO will probably never see it, it will go straight to one of his minions who will pass the buck "Nowt to do with me", and pass it on again,and so it will all go around in circles.
Airlines, because of the business they are in, have legal teams that know the industry inside out, the rules and regulations are unbelievable, and are also eye wateringly expensive to comply with, they can't afford to get it wrong, and they only see the customer as a pound sign. They will fight any claim for damages with vengeance, as they only make about £5 per passenger profit on each flight, that means that they stand to loose about one full Boeing 757 flight sector revenue if they have to pay for your bike.
If you do try to contact the airline direct, then address the "Station Manager" of the airline at the airport in question it is he/she who deals with the daily runnings of things, make sure you put the date it happened on your letter, so he/she can speak to the "duty manager" of that day, as they work shifts
Your travel insurance will be your best (and easiest) bet, but they may want pre damage pictures etc.
Have you spoken to ABTA? they may be able to give you better advice, and tell you what to write/say to them.
Unfortunately with budget and and holiday charter flights, you are cattle, get them on, get them there, empty it, fill it up again and get them home all as quickly as possible, and so the cycle goes (no pun intended), customer service only extends to getting you on the correct coach at your destination, and you will be lucky to see anyone at all at your UK airport unless it is the sales desk.
I went to Spain about 5 years ago with BA, and my golf trolley got damaged on the return flight. They dealt with it straight away at arrivals in Manchester, i filled out a form, and within a week a courier arrived at my door with a replacement, no fuss, no ifs or buts. The flight wasn't that much dearer than a budget one either, and i got the use of the BA departure lounge at Manchester, much more civilised.
Good luck, i am on your side. Keep us informed, i admire anyone who can take these pariahs on and win.
Jet2 are owned by Dart Aviation, they may take more interest in what you have to say, try dropping them an email as well, you never know a 2 pronged attack might work.
