I'd suggest ripping out as much of it as you can find, then laying down sheets of cardboard (or newspaper in layers at least four sheets' thick) over the ground. Overlap the sheets so there are no gaps between them. Top the cardboard/newspaper with at least two inches of mulch, and leave it for a few months. Any creeping Jenny that still emerges should be pulled ASAP, or spot-treated with Round-Up by painting/wiping it directly on the leaves (avoid spraying which might travel to plants you want to keep). In time (if you're in a wet climate), the cardboard/newspaper will break down, but by then, the creeping Jenny will have been smothered to death. Additionally, any weed seeds falling onto the mulch will be prevented from getting to the soil by the cardboard/newspaper, and can be pulled very easily if they sprout. Once the creeping Jenny is gone, consider other short-growing plants to fill the ground. Low-growing Sedums were mentioned, but there are many others. Consider Prunella, Thymus, various short-growing Campanula and Geranium, Dianthus, Lithodora, Lamium, low-growing Nepeta or Salvia or Veronica, etc. Also consider self-seeding annuals for this "job" -- their root systems won't become as substantial as perennials, and if you don't like them or if they start spreading too much from year to year, just keep them from flowering/seeding and there won't be a new generation. :-) ~Christopher

I have a high deck too, and I don't have anything planted under there. I would be afraid to make it hospitable to snakes and other type critters. I put a few stepping stones in the ground just to make paths so I could get to things: I store plant stakes, empty pots, and other such important but nasty looking stuff under there. The only thing that grows under the deck are a few seeded sweet gum trees and such that grow near the very edge (and the runaway Ruellia that I can't seem to eradicate) but they all stay close to the edges where there's a bit of light. I try to handpull or round them up occasionally. Unless you have some good reason (and I would like to know if there's something we're missing), I would leave well enough alone.

Ajugas, Ceratostigma plumbaginoides, Lamiums, Lysimachia nummularia aurea, Geranium maccrorhizum or related hybrids like 'Biokovo', 'Karmina', etc. These come immediately to mind, but there are others.

If you're willing to water it to get it established, you could try corsican mint (I know, mint loves sun, but this one thrives in deep shade) It would not interfere wiht the existing air circulation, an very important consideration to avoid rotting out the porch, and would smell great when Rover stepped on it. He'd smell good too. You could install plugs of it next to the lattice, don't kn ow about seeding it though. Moss might work too, find a clump of it in deep shade in your yard and move a little to the porch, water, and see what happens.