If your plan is to keep the existing flower garden only, I would mow the weeds with my trusty lawn tractor. I'd overseed with grass seed in the fall (or whenever is the best time in your area), apply a pre-emergent weed control in spring, and a weed and feed in the early summer, carefully avoiding the flowers and any other ornamentals, as the weed killer will kill them. Hand weed a strip near the flower bed. Overseed again next fall, repeat until you have a decent lawn.
Once you have the weeds more or less under control in the strip adjacent to the flower bed, plant a shrub border as a backdrop for the flowers. Evergreens are nice for year round structure, or use whatever is most reasonably priced if cost is an issue. You can even start some shrubs from cuttings now in containers, in anticipation of needing a hedge.
Many shrubs are relatively easy to start from cuttings, and you may know someone who has a suitable plant that will give you some.
If that seems like too long a process, a simple post and rail fence behind the flowers might be nice if it suited your house style.
Or a simple homemade trellis with something grown on it. Or you could just treat the bed like an island planting and mow on all sides.
In all honesty, I'd probably end up planting the whole thing with shrubs, trees and flowers, but that wasn't what you asked. :oP

Jo

Mow the area down; spray with 'Burnout' or 'Scythe' or other organic weed killer; wait a week and spray again;next lay down black plastic held in place with rocks to smother weeds for a month or so. Remove plastic; lay down weed cloth; mulch heavily with bark mulch or gravel. That should do it, hopefully, unless weeds are primarily of the crabgrass and dock ilk!