Someone stepped on my begonia a while ago and a big piece broke off so I put it in water and it’s still alive and flowering (flowers changed colour recently but that’s another story). Anyway they’ve been in that glass with water and a bit of earth for weeks now and there’s not sign of roots… yet the piece is obviously not dying.
What should I do? Can I just stick it back in my garden and water it a lot?
Pick those flowers off. Those will inhibit rooting. Also, I recommend crushing up an uncoated aspirin tablet into some warm water, and stirring as well as possible. Let this cool, and water it with the mix. This will force rooting much faster.