Getting Rid Of Wild Grape Vines?

We have a flower bed that has been sorely neglected. Elm seedlings and wild grapes have invaded. There are still some desirable plants - some Japanese dwarf iris that haven't bloomed for a couple of years and an antique rose that was very nearly choked out by elms, grape hyacinth, and some yellow-flowering bulbs that I haven't identified yet.

Because of a bad back, I can't spend a lot of time weeding at once. How do I get rid of the wild grapevines and baby elms without harming the other plants? Or should I consider it a total loss and till the whole thing up?

Hardiness Zone: 6a