In ASIO mode, you can only use 1 device at a time, so yes you'll have to mix through the UX2.
Just get an adapter (if necessary) so you can plug your computer speakers into the UX2.

57Gregy
In ASIO mode, you can only use 1 device at a time, so yes you'll have to mix through the UX2.
Just get an adapter (if necessary) so you can plug your computer speakers into the UX2.

Yes, you simply need an adapter that breaks out your mini plug into two 1/4" jacks. The UX2 output will probably produce better fidelity than your computer's speaker outputs anyway.

Alternatively, if your internal soundcard has a digital SPDIF input, then connect this to the UX2's SPDIF output.
 

Thanks Craig, coming from you that means a lot to me. After all I've known you for 30 year or more, but you just met me :)
Most of what I know was learned by reading magazines and books on audio, so your one of my original Professors of Tech! 
 
JohnnyB,  As a few have said, you really don't want to have to switch drivers just for playback of your audio. You will still have issues, You could even revert to your on board sound cards drivers which probably won't interfere with line 6, but this is still a funky set up. You audio interface can and will do everything for you and do it better. 
 
What I have done when travelling and I have my interface but  I don't have my studio monitors.
If the computer I'm using has typical desktop speakers I plug them into the headphone jack of the interface. All you need is the 1/8" to 1/4"  TRS adapter that comes with most headphones.  
This is actually what computer speakers are made to plug into- a headphone jack.