I'm tempted to pull up the original bed, as well as all the rogue lilypads, then replace it with some coleus or hostas. But I fear we've already lost the war. Is getting rid of this stuff impossible?

big lilly pad looking leaves sounds more like my local wild violets to me...

but both of them are indeed invasive...very invasive, and it takes a fair bit of dedicated pulling (once a week for the rest of the season, and likely on into next year) to get rid of it.

It is a biennial and develops a deep tap root. Additionally, it developes cute little pods that look a bit like a tiny, long, thin string bean, which throw numerous seeds all over when they ripen and twist open. If you crush some of the leaves and smell it, it has a distinctive aroma which is unlike lamiastrum.

I think the birds really like this stuff and seed it all over the place.

You are, however, right to keep an eye on your Yellow Archangel (ie Lamiastrum). Some web entries confuse it with Herman's Pride, which is pointier looking and reasonably well behaved.

My Yellow Archangel is spreading all over a dry, deep shade area in my backyard, and I am considering rescuing the few good plants (Solomon's Seal, a few ferns, ginger) and letting it do its darndest. I know it will not let the wild mustard back in, and I'm tired of fighting that noxious pest (even though I think I read somewhere it could be edible...?)

I used good old Y.A. in some pots out front, and by the end of the season it had managed to establish itself in the areas next to both pots. It is presently duking it out with English Ivy in one spot, and I think it is winning.

I had a scare earlier this season when I noticed some similarly-shaped leaves coming up yards away, but without white markings. I was relieved to discover that those leaves were seedlings of a scullcap plant I had planted last year.

The main point I want to make is that this 'wild mustard' is called garlic mustard. Its one of the worst invasive species in woodlands in the midwest. It is not desired or spread by birds, the seeds kind of catapult themselves out from the parent plant, and otherwise are spread mostly by mammals (mostly deer, humans and squirrels) who track them on their feet and occasionally in fur.

I would recommend eliminating both of these species from any location in North America. See the attached link for more information about garlic mustard.