How "small" are your koi? What is your location? What has your temperatures been like during the day and is it in full sun?

koi are about 5 inches in length.
I'm in Southern California.
This past week has been mid 70's with sun starting at around 1pm, but two weeks prior to that temps have been in the 80's with full sun. It is not in full sun, but it probably gets at least 8 hrs a day.

This looks like normal pad deterioration to me. Pads come up and they only last for awhile and die off, replaced by new ones. If you are not fertilizing it though, you need to be doing so. Lilies are heavy feeders. You can get pond plant tabs or if you can find them in your area you can use Jobe's Tomato Spikes. Follow the directions on the plant tabs or use 1-2 tomato spikes every 30-45 days during the lilies growth season. Your koi are not the problem. Koi will root around and dig in lily pots but I don't believe that is your problem here.

Thanks!

When I clicked to enlarge it, it also appears to be a Tropical Waterlily, not a dwarf hardy. There aren't any dwarf Tropicals though Dauben will adapt to a smaller pond than most.

Looks like normal deterioration to me as well.

Yep, I think you are right, that looks like the pads of a trop.:)