Coming from a strictly rational perspective, which I figure many users here would appreciate, there is no real clear way to
know if one is "fixable," as you say. In order to answer the question, we have to be very specific about what fixable
means. When you really dissect the idea, you will find it is difficult to pin down a definition that we are satisfied with.
Someone who's not thinking dark depressive thoughts 24/7 and is happy/content and resilient.
For example, using your definition here and speaking rationally, I see no reason why this isn't an achievable goal. Let's say you get yourself to feel happy/content and resilient for just a blip of your day. That is a success! Lets say you can get this feeling for one hour a day, are you now fixed? Perhaps then you will consider other parts of yourself to "fix" and you may never really come to a point mentally where you accept yourself as you are. In that respect, there are people that are not fixable.
Some people are not fixable, and other people are. I would argue that all of us are combinations of both. We all have self-destructive thoughts of various degrees.