Former bank loan officer speaking-
A broker isn't always the best route to go, largely because they are just shopping for you. They may not have actually found a bank or lender yet to take your loan. And to ensure they keep your business, they're going to keep you on a string waiting...
If you have excellent credit, no money issues, and enough value in your home to make a refinance doable...go to a bank. Go to the bank you bank with or walk into a bank if yours doesn't offer what you need. You might get a too good to be true rate through a broker, but could pay the price in other ways...time, hidden costs, limitations on what you can do with the home, etc.
There is no reason approval should take 2 months...approval takes about 10 minutes! Even if a bank if backed up....you should know within a week or so of application. Now, appraisal and closing may take longer, particularly if you have to close through an attorney, but approval is fast. Stop allowing this person to waste your time and potentially your money. Sit down with a loan officer, speak face to face if at all possible and make sure it's worth it to do this in your particular situation. A refinance often isn't the best route, but a good loan officer will take you through the proper steps to reach your financial goal, be it a lower rate, lower payment, or a way to free up cash.
I wish you luck!