I think there are some lamium varieties that are less invasive than others, but I had some of the common yellow-flowered one that has driven me crazy for years. It pops up everywhere and is hard to keep contained.

you could try a saxifrage, (like Saxifrage x geum) but I'd guess that the sweet woodruff will spread and fill in around any other groundcover you add.
other ideas include ophiopogon, or a lirope.
various primroses will work well for an edge, or perhaps Geranium macrorrhizum, or lady's mantle - Alchemilla mollis.

Is the area dry or wet?

Beesia, Soldanella, Asarum, evergreen Pologonatum, ajugas,....................etc.

Check out Leptinsella squalida "brass buttons". Grows in sun or shade. Forms a thick matt you can walk on but is slow growing, not aggressive.

I find Lamium to swamp other plants and I'm sorry I planted it in my mixed shade bed. Ditto Sweet Woodruff. I planted the pretty purple and pink bloomers with gorgeous foliage to accent hostas, ferns, dicentras and huecheras and it grows right over the top of them instead of in between.

I was able to get all the brass buttons out a few years ago in another bed where it spread to where I didn't want it but didn't grow where I did.

I don't mean to sound negative, but I've not found a neat & tidy edging plant for sun or shade yet. A stay put less 6" wide and high edge seems best installed as rock, concrete, brick, etc. There are numerous products available.

As long as I understand the growth habit of a plant and plan for it in the bed a lot of different things work especially when I cut back after flowering to keep them in check and foliage good looking.

-English primrose Wanda dark purple with yellow eye early bloomer is my best edger for next to a path. I dig & divide a section at a time both in spring & fall, so that over 2 years I've done most of it. If I don't it crowds itself out and black vine weevils serrate the leaves. I also cut it back whenever it needs it usually after flowering, mid summer after a hot spell, and again after flowering in fall or can wait until late winter when looking pretty bedraggled and new foliage is emerging from the crown.

-lamium is going to spread 18" wide and flop a bit

-Georgia Blue veronica needs 2'

-sweet woodruff spreads in whatever space it can

-hardy geraniums need 18-24inches

-lady's mantle mixes well, but flops forward 2-3' when blooming

-Germander (Teucrium chamaedrys) takes clipping to keep it neat, but is evergreen with shiny leaves growing best in sun with good drainage. I'm still working on getting mine to fill in & edge an herb garden. It might take more care than I give it to fill out better.

Some annuals might work if you're willing to feed, water, & replant each growing season.

Maybe you'd like 'Ice Dance' carex morrowi. I have it lining a path in a shady area and like it a lot.

Here is a link that might be useful: 'Ice Dance'

This was the weekly tip in Horticulture Smart Gardening email today. I hope it helps you decide what you'd like to use.

Here is a link that might be useful: Define Your Garden with Edgings

Thanks everyone for the suggestions. Corrine, I don't know why I didn't think of a stone border. Would solve a lot of problems (and expense)and I think it would look great. I followed the link you gave and found 'The Flower Gardener's Bible' on Half.com for $1.99

So glad that helped.

After visiting perennial gardens at Manito park in Spokane this weekend and returning home to tonight I realized how much I do prefer an edge whether stone, brick, or low growing plant. The public gardens there had a cut edge and some had some lower growing plants next to the edge, but none were edged with plants.

At home one bed of mine caught my eye tonight with flowering Serbian bellflower in part shade. The leaves are toothed, but in a frilly way and so showy in bloom. Though last spring I took out a lot of it as it had tromped all over other plants. As long as I space them 12" apart and keep other smaller plants away from it I am happy with the look. I also cut it back hard after flowering and whenever the trailing stems don't stay in the garden. It is evergreen and faithful to return year after year. It divides easily & because of its increasing habit is often brought to the Green Elephant plant swaps in Redmond several times a year.