here is how i would set this up...

1. ipad audio into mixer..
2. performers microphone into mixer..
3. audio from the ipad routed out of mixer to room speakers using a monitor send or aux send on the mixing desk

4. main outputs of the mixer sent to the cameras audio inputs with correct balance set between between ipad and microphone
5. camera is captured into vMix (audio and video)

6. if you need videos playing from vMix to be heard in the room also.. route the video clips to an audio buss in vMix and output that buss from computer interface into the mixer. then you can raise the vMix level in the aux send of the mixer to the room speakers..

if have audio from ipad running through speakers and a mic in the room it will be difficult to totally avoid echo.. you could maybe try using headphones for the performer or point speakers towards the performer from other side of the room and use a cardioid microphone..

a benefit of sending all audio sources to the analog mixer is you have everything in one place and can then do one mix to the speakers using an aux send and a separate mix back to the camera for the stream.. (with this setup all audio for the stream is coming directly from the camera)

that isnt the cleanest signal path as the video clip audio goes out of computer, through analog mixer and camera back to vMix again. the benefit though is it helps avoid latency and should be easy to make adjustments to speakers or stream levels from one location.. if you want the images from the video clip to be in time with the performer singing you may have to apply some delay offset to the audio on the video clip before sending it out to the mixer..

as an alternative... if your audio interface has an internal mixer and at least a couple of inputs/outputs you may be able to connect directly to it instead of the analog mixer for a cleaner setup.. ie:

1. microphone and ipad go directly into audio interface.
2. add 2x audio inputs in vMix for the microphone and ipad
3. send the video clips audio to a vMix buss
4. use audio interface mixer software to adjust levels between microphone, ipad and the buss from vMix with the video clip audio. (reliant on the interface being used offering this feature)

5. use an output on the audio interface to connect to speakers and send the monitor mix created in the audio interface mixer.
6. use vMix to set levels between microphone, ipad and video clip for the main stream audio
7. adjust delay offsets in vMix as needed for different sources..

i cant picture in my mind exactly what delays would be needed using this setup but should be able to get everything aligned with some tweaking (i think).... :)