Interacting with the touchscreen of a mobile phone in virtual reality (VR) is challenging because users cannot see their fingers when aiming for targets. We propose using two mirrors reflecting the front camera of the phone and a purpose-built deep neural network to infer the 3D position of fingertips above the screen. Network training is self-supervised after only a few hundred initial labelled images and does not require any external sensor. The inferred fingertip positions can be used to control different hand models and objects in VR. Controlled experiments evaluate tracking performance for single-finger touch input, and compare several 3D hand representations with a flat 2D overlay used in previous work. The results confirm the suitability of our fingertip tracker to aid precise tapping of small targets on the phone screen and provide insights about the effect of various hand representations on control and presence. Finally, we provide several application examples showing how 3D fingertip input can complement and extend phone-based touch interaction in VR.