I see 4 options:

1. If there is some container (bath wash, pan), he upends it and use the air inside once the bath is full.

2. If "at this rate" means the current speed (litters per second), then he have more time, since when water surface will go up the water pressure will increase and compensate pressure in the bath faucet knob, therefore the speed will decrease with time, which can easily give him additional 5 minutes.

3. If he has clothes or towel inside he can use it to plug the faucet hole, this will decrease the speed.

4. If he has phone he can call someone to help.

P.S. If the water pressure is big enough there is no way to escape: he even can be smashed by it ones the bathroom is full.