You can travel from Turkey to Arbil for a good price by this budget airlines:
www.flypegs.com with stop over in Ankara it costs about 131 UK pounds
or www.atlasjet.com direct about 240 US dollars
Its Only 1:20 hour:minutes flying.
I have seen that Kurdistan Airlines flights from London and Birmingham. They don't seem to have a website but a travel agent has details on their website. http://asiantravel.eu/ I can't vouch for them and have no idea if its real but it might be worth a try.