I have a Corda HA-2 that does an excellent job with both my DT880 and k501 cans. The HA-1 Mk II has specifications similar to those of the HA-2 (in terms of its output voltage and current) and should provide similar performance (although I have no first-hand experience with that particular model). The lack of a 120 ohm output jack in the HA-1 Mk II is of no consequence, since both these headphones sound best with a low impedance output.
That said, the key to making the k501s sing is getting enough current into them. Many otherwise fine amplifiers fall short in that department and the k501 has acquired an undeservedly poor reputation as a consequence. I will admit that my k501s sound even better when driven by my Grace 901, but that's a much more expensive amplifier with an output current capability roughly double that of the Corda amps mentioned above. It's hardly a fair comparison but it does serve to emphasize the importance of current.
Another possibility might be a PPA with plenty of output buffers and a hefty power supply. I'll leave it to others to elaborate on that theme as I have not heard a k501/PPA combination. Just speculating.
I trust that you have auditioned both headphone models you are considering, as they have radically different sound signatures. There are plenty of threads that pontificate on their relative merits. Without knowing what type of music you listen to most, I would not dare to make a personal recommendation but I would venture that the DT880 is the easier to drive and, therefore, could potentially increase your amplifier choices. If you're leaning toward the Corda, you won't go far wrong with either set of cans; if you can find a used HA-1 Mk II at a good price, perhaps you could afford both headphones and not have to make a difficult choice!
