Looking it up, it seems the Asus Rampage III uses a Realtek ALC889, which if implemented decently (and with the drivers working right) actually is a very usable affair. A run of RMAA to check for resampling (make sure sample rate is set up correctly in newer versions of Windows) and usage with some more sensitive headphones to check for fuzz and buzz should be sufficient to get a reasonably good impression of how it performs. If it does well, chances are that audible differences between this and a "real" soundcard won't be present.
K240S' (much like 50 ohm HD595/598s) are a little fussy when it comes to amplifier output impedance - you'll want 10 ohms at most, with less being desirable. Most serious amps will be at <= 10 ohms, but you have to look a little harder for something that does <= 1 or 2 ohms. I know for sure that Jan Meier's Corda Swing (*) is one of them (along with Lake People amps, RSA's Hornet, iBasso T4 and P4, and the iBasso D12 is at a fairly low ~2 ohms still so only critical multi-driver IEMs will be bothered). Not sure about FiiO's E9, it's fairly different from its portable cousins E5 and E7 (which do have very low-impedance outputs).
(*) Not to be replaced once stock runs out, sadly.
Soundcard wise, you usually won't get terribly wrong with one of Asus' own internal offerings, assuming you don't get any driver issues. A DX or D2X is a fine card with excellent parts quality. (I don't quite remember which feature the D1/DX didn't have, I think it was DDL.)
