Having a weed grass invade a planting area is often hard to deal with. Often the weed roots (stolons) will invade the root system of the desired plants, complicating removal.

If this were a planting bed with some other sort of groundcover, you could use a herbicide that targets grasses/monocots only, like Grass-B-Gone. But that won't work with the mondo grass, as the herbicide can't distinguish between it and the weed grass, even though the mondo is not a true grass (but IS a monocot). Any other type of weed killer/herbicide would damage anything in the bed.

I think you are limited to manual removal, unfortunately. That may involve actually digging up some of the mondo in order to affect a complete removal. But it can be replanted easily enough once the bed is clear of weeds.

A barrier/border should be helpful after you have cleared unwanted plants from the area, so the creeping grass can't creep back in. Mow with chute aimed back toward the mowing area, not toward beds, if there is any sign of seeds on the grass while mowing.