I've used the following analogy:
Imagine if, one day, you lose both your legs. Chopped off, right at the hip. You can no longer walk, no longer run, no longer dance. You can still function relatively normally in society, but you feel restricted all the time - no longer quite free. Furthermore, you still feel like your legs are there - you can sometimes sense them, you expect to see them there, in every sense you feel like you have legs, and should be able to run, and dance, and do all those things freely. But you can't. You wake up in the morning, and swing your legs over the side of your bed... and they're gone. And for just a moment, you can't remember why. When you go down the street in your wheelchair, people stare at you - you no longer quite fit in. You're now something subtly other to them. But you still feel like a legged person, feel like one of them, and you still expect others to treat you that way. And whenever you move down the street, you're acutely aware of just what you've lost, and how other people now look at you differently.
But even worse is, you could be like them - you could get prosthetics, good enough to walk, if not run, and good enough that you no longer find yourself surprised every time you look down. You might even be able to get transplants. But the people around you won't let you. They don't believe you when you say you used to have legs. They can't remember. Even your parents refuse - "honey, you were born without legs. You just need to learn to live with it."
And the worst part is, they're right. You never did have legs. You just feel like you did. And they refuse to treat you, to give you new legs, because they refuse to accept that you want legs. Because, surely, others have gotten used to it, and having lived without them for so long, so should you? But it doesn't work that way, because the only thing that matters to you is being able to dance, and run, and climb a mountain. Without those, life isn't worth living, even if you can live mostly normally. Because what lies between "mostly" and "normally" is the most important stuff of all.
Now, imagine that this was pretty much the only thing you could think about.
It's not perfect, but I think it works.
I've never had much of the mirror problem, myself. Seeing myself naked is bad, because every time I see my body with that thing hanging down there, I'm surprised, unless I consciously braced for it. But my face and body are otherwise feminine enough that, most of the time, I can see a woman looking back. Sometimes, the male image snaps back into focus, and then I want to die, but most of the time, I can make myself see only a slightly ugly, macho woman.