The problem is that the mixer is fully monaural (mono) and your headphones are stereo. Your mixer’s output is not designed to work with headphones. A TRS cable connecting the drum module to the mixer will not help because the mixer will only see mono.
The quickest, easiest way to play along to music on your iPhone is to plug the iPhone into the drum module's AUX IN jack (using whatever adapter you're currently using to plug it into the mixer) and plug your headphones into the drum module's PHONES jack. Then you'll hear everything in both ears and in stereo. This use case is what the AUX IN jack is for. You don't need the mixer at all.
In comments, you asked if you’d be better off using the MIDI OUT. The answer is no. MIDI is a digital interface that carries performance and configuration data, not audio. (It allows electronic instruments and computers to control one another.)
If you were dead set on using the mixer, you'd need to buy a headphone amplifier that accepts mono input and is capable of sending the mono signal to both ears. And, even then, you'd only hear one side of the iPhone's stereo. On some familiar recordings, you'd definitely hear things missing.
There are a few other points to make that don’t apply to what you’re trying to accomplish, but may help you understand your gear more fully.
To connect your drum module to a mono system, you would ideally use the L/MONO output, not the PHONES output, which is designed for headphones.
An additional problem with using the PHONES output to feed a mono system is that the individual pieces of your kit will have uneven volume. In the stereo field that you hear from the drum module’s PHONES output (when you plug headphones into it), you’ll hear hi-hats more strongly on one side and ride on the other. Same with the toms. If you connect this stereo output to a mono input, you’ll only hear one side of the stereo field, causing one side of the kit to sound weaker than the other.
To connect your drum module to a stereo system, you would want to use both the R and L/MONO outputs, not the PHONES output.