It's taking over three months for the application to go through then another month after the CAA rip you off for 44 quid until you get your FAA letter to apply. So you need to apply tomorrow if you have any chance of getting it sorted before you go over there!
Holiday insurance doesn't cover the flying, you will need renters insurance. Application on AOPA site & you will need an address in USA for the insurance.
Been there seen it done it!