D.B.
I think your role is to support her, and perhaps to offer to attend an appointment or two with her. It can be extremely helpful to have a second set of ears and someone to take notes.
I think she has to go well beyond GPs and ENTs for weight gain and vertigo - seeing an ENT makes sense, but she didn't get answers. I would urge her to call a major teaching hospital with multiple top specialties under one roof, and get an evaluation under various disciplines with all of the specialists talking to each other and reading each other's notes. My husband just had a very rare condition that took a long time to diagnose - he had symptoms of certain things, but not all that you would expect - so we had to keep digging for answers. We had a phenomenal experience at a major Boston teaching hospital and I can't say enough about how extraordinary the partnership was.
I think she may need another look at her thyroid issues - I know plenty of people who've had their thyroids removed for one reason or another, and just because they are on medication doesn't mean their problems go away. And I'm not sure she's got the right caloric intake and the right balance of nutrients - whether she's not keeping track or whether she's using any reduced calorie sweeteners, for example (even the natural ones which often have other things mixed in during processing). But there may well be other things no one has considered, and she needs some people to think outside the box in a collaborative way.