How safe is having my boiler in my daughter's bedroom?
Gas leak
A gas leak could be deadly not only due to inhalation, but also as an explosion hazard.
Carbon Monoxide
Carbon monoxide is an silent killer. It's colorless, odorless, tasteless, and can kill you before you know you're dead.
Combustion Air
If the home is sealed tightly, and a source of makeup air is not present (for example if the source is supposed to be the open door, but the door is closed) there could be a suffocation hazard. As the boiler burns fuel, it also uses up air (oxygen). If a sufficient source of new air is not available, the life supporting oxygen in the room could be used up.
Fire
Any time there is fire (or any heat source) in a home, there is always the potential for an uncontrolled fire.  A fire starting so close to a sleeping child (or anybody for that matter), could likely be fatal.

I would not put my daughter in a room with a fuel burning appliance, or any appliance for that matter. Utility appliances belong in a utility room, not a bedroom.
NOTE: I wouldn't put my son in the room either, if I had one.  I may put an in-law, or a house guest that's been around too long in there though.