I start with a few questions.
1. How much of the $13K will be left after you pay the closing cost for the refinance?
2. Can you drop your rate by at least 2%?
3. How long will it take you to gain back the equity you are using and will it be worth it in the long run, especially since you will have anywhere from 3 to 5% closing cost to refi, depending upon the lenders cost or if you can get 0/0; meaning no origination fee and no discount fee.?
4. How long do you plan on staying in this home?
5. What type of loan do you have? Conventional/FHA?
A cash-out refinance will cost you more. By the time you pay the closing or add it back into the loan (do not know your balance); you may not have much $$$'s left. Think about the questions above. Yes, the rates are lower than 6.125, but when considering how many years it will take you to get back where you are in equity; is it worth it at this time? In my underwriting and origination experience; we did not recommend refinaning unless you could drop the rate by 2%. It is really not a big difference in payment and especially adding back the closing cost and only getting minimal funds which would be insufficient to do your renovation-add on.
My thoughts are that if you really want to add on to your home; you can possibly get a bank (construction) loan to do the add on, based upon the completed value. (I do not know who you bank with; and since the economy is like it is; I do not know if all banks do this any more, but has been common practice). Just like the banks do a new home construction, have the work done and then refinance the first and second with a new loan with the new value. You would have to have some kind of house plan for the add on, the bank would appraise it etc. You would benefit more from this method and have less closing cost because the banks closing cost for the construction loan would be nothing like the first refinance you are describing and then have only one mortgage refinance closing.
The aforemented could save you closing cost for one loan and I would not want to touch the annuity fund...that's just me.
I hope this helps and good luck.