First two things from "email...area manager"
In my company my middle manager would not appreciate being bypassed. Is there a more direct manager you need to approach first, even if they just refer you to the area manager?
Secondly, email isn't ideal for this. I'd prefer a face to face meeting, if that can't be done then at least start with a phone call and follow up with an email.
Contents: I don't think I'd talk about offers from recruiters/market rate etc. Negative approach and could be perceived as a threat. Manager can easily turn round and say, "if somebody else will pay more then go and get it". I would focus on a positive approach, explain the value you add in your role hence why you are worth more. If I were going to do it I don't think it would just be for 10%, that's a rounding error really, if you're going to ask I'd be looking for at least 20% (though you may not get it). I'd avoid mentioning exact numbers though, better if the other side mentions figures first.
With your situation of, by the sounds of it, being asked to bail out another failing team for a couple of months, I'd consider making the initial conversation about a set of targets for you agreed with manager for the two months assignment, a chance for you to demonstrate value added, which if achieved would then justify a conversation about a salary increase. A bit more of a message to the manager that you're going to solve a business problem for him, and then want commensurate compensation in return. But most likely you have to do your bit first without a guarantee of the salary increase.