Sounds reasonable, but asking for double what I'm already making seems ridiculous.

It sounds to me like you're not so much unsure of how much your position is worth, but the fact that it's so much more than you're making now that you're hesitant to really say. For one thing, I'm sure your bosses recognized even at the time that they were underpaying you. But now that they see they'd like to keep you, they started to worry about what would happen if you got another job offer. If you got one, then they'd have to agree to give you a raise to keep you, but that would make them look bad. Doing it before that happens makes them look like the good guys, but know that they're not being selfless.

Also, they've put you in a position where it feels like you're asking for a raise. But they haven't. They're just asking you what you feel you'd like to be making. I imagine that the worst they'd do is say no. Your job is secure; it's unlikely that they'll let you go because you ask for more than they were expecting. At the same time, don't ask for a million-dollar raise.

Be honest with them. Tell them that you didn't and don't know a lot about such things, especially since you'd been out of work for quite a while beforehand. Tell them that you looked around to find out what comparable positions pay and came to the conclusion that it was x. Say you feel that you think that you're above average at your job, something that would seem to be corroborated by the fact that they want to give you a raise, and that a salary should reflect that, so that would bring it to x + y%. Then say ``at the same time, I recognize that that's z% more than I'm making now, and I can't really expect that kind of raise, but you asked me, and that's what my research shows''.

Keep in mind that the fact that they've offered you a raise means that they want to keep you.