Default Phone insanely slow

My daughters HTC Desire (not rooted) seems to have given up.

When I boot it just rendering the "Quietly brilliant" takes around 10-15 seconds. And then after several minutes the boot up sound is played. Then a while later the battery is displayed and the "Connect your charger" text. Even though the charger is connected. I cannot swipe down to unlock the phone as it just sits there. Eventually rebooting.

I have managed to get into recovery but trying to do a factory reset there has left the phone displaying the blue/purple recovery screen for two whole days without doing anything. Just clearing the cache through recovery took around four hours.

Has anyone seen anything similar? Or can the phone just as well be thrown in the bin?