The main factor to take into consideration is that the headphones determine most of the sound, and even then, the differences are not always groundbreaking.
Going from the HE400i (mine was the original version, but with a modified grill) driven from the Schiit Fulla (a combo that was originally under $600 but is now barely over $300) to the HE1000 (granted, I have not heard the HE1000 V2) driven from the Simaudio MOON Neo 430HAD (a combo that exceeded $7,000), the improvements were not large, in my experience. It was like an enhanced version of the same basic sound signature. Diminishing returns is a brutal phenomenon. (I always make sure to test with all the different types of music I can, by the way.)
Due to this and many other experiences, I'm more than a little skeptical of the objective benefits of high-end amps. The Jotunheim (amp without the DAC add-on) is not that expensive, though. I'm not sure what you mean by "$500 better than the Lyr 2" because it costs $399, whereas the Lyr 2 + LISST is over $500. Since you already have the Lyr 2, you certainly don't need to switch to something else. It already can output more than the Jotunheim at 32 ohms. (6W vs 5W...though you're never going to use anywhere near that much power unless you crank the volume to max.) So it comes down to the sound signatures of the amps. I can't help you there, since I have heard neither.
As for DACs, the most expensive one I've owned is the Chord 2Qute (originally $1,795 in the US) and the cheapest one I've owned (not counting onboard DACs of computers) is actually the one I'm using at the moment: the $30 Behringer UCA202 audio interface. I believe that DAC upgrades are worthwhile, but definitely not necessary. There's no guarantee that you will even hear a difference, though I do think it's likely. I think the Schiit Modi Multibit is a good DAC that isn't too costly.
