I bought the AP last year just for the discount. We had an extended stay, 12/14 - 12/26. I don't remember all the of the specific calculations, but I can tell you that I know the pass basically paid for itself. We always stay at either the BC or BW. The original reservation I had was without the AP discount. I bought the pass, and then switched the room to the AP room only rate. The room discount paid for my annual park pass. I did not buy the kids annual passes, which I regret. (As we have since returned.) In addition, I recently used the pass for a room only reservation at the BW for 1 night of our week long trip. The AP allows one to purchase the DDE card. A 20% discount for most meals we ate.
One footnote though, I said we wouldn't go back for at least a year....I found myself saying "well, I do have an annual pass..." Did I really save anything? No. We just vacationed again. I think that's Disney's philosophy and it worked on me.
If you're more disciplined than me, it can be very cost effective.
One other thing, a room discount offering is not a guarantee, and I had to check daily to get 8 consecutive days at the Beach Club Villas. (for about 2 weeks!) Also, because I checked in prior to the holiday rate, the value rate applied through my stay. It is my understanding that practice is ceasing next year.
Because of your extended stay and the fact that the rate will stay the same through the holidays, it is probably cost effective for you. Especially if you plan to visit the parks more than 10 days.