The FiiO E11 should be fine for this purpose.
"Intel HD audio" does not really tell exactly what you have, as it is the actual (most commonly Realtek) codec chip and its implementation on the motherboard that matters. I have an ASUS motherboard with Realtek ALC887, so it may or may not be relevant for your machine, but I found that:
- the rear jack is a line output only, and not really suitable for driving headphones (200+ Ohm output impedance, small coupling capacitors)
- the front panel headphone jack has some amplification, but it can only output a maximum of about 1.2 Vrms, and has an output impedance of 77 Ohm (of which 75 was probably added by ASUS, and only 2 is in the Realtek chip according to its data sheet)
- without modifying the front panel, there is audible noise and interference even with the DT770 Pro 250. I was able to fix this by making some changes to the front panel to eliminate ground loops
- the Realtek chip has digital volume control, so reducing the volume does not reduce the absolute level of noise
It does not actually sound that bad (other than the above mentioned noise issue), but it does not allow for really loud sound; I find it enough for music with low dynamic range, but your mileage may vary. The maximum power is higher than with a decent portable player, but not that much higher. The 80 Ohm version would be somewhat louder, but also make the problems related to noise and high output impedance worse. In any case, I think the combination of Xonar DG + 250 Ohm is likely to be better than onboard + 80 Ohm.