Alternately, you can also get a music server like the Musical Fidelity M1 Clic and the Aune S1, and hook up their RJ45 terminals to that other router. Having that access point router makes such a set-up easier, btw, because you don't have to run the cable to the primary router, and as opposed to using a DAC, these two have quality output.* The S1 already has a headphone amplifier output also, just doesn't have its own knob; not really important to most people but I prefer knobs for quicker and more controlled volume changes, but I use my DAC-Amp with a variety of material (web content on top of my own music), so if it's just your own music and you equalize gain when you rip, it's not really a problem.
From there you can get the appropriate app for your iOS or Android device to use them as a remote control and see the content on these, instead of using the built-in screen. Usually this is only really necessary for speaker systems where you're farther out from the screen, but depending on the quality of the remote app, it can still be smoother to use than the buttons on the device itself or the remote it might come with considering the loads of content you'd have access to.
*Of course, I wouldn't expect the M1 Clic to be identical in output with DACs and CDPs at the same price point; the S1 probably, because it's a cheaper brand and product, compared to a $600 CDP. Then again, there are $1000 CDPs I didn't really like, so that's not the sole determinant.