If it were my garden, I would dig out now whatever plants I want to save, dig out the grass and weeds, apply a 4" layer of compost, mix it in, replant the flowers/plants and apply a 2" layer of mulch.* You can make a drop down trench edging.* It's near the bottom of this page under 'dropped lawn edge' here.
Using landscape fabric does not allow the worms and other organisms to mix the soil for you, including the mulch as it decomposes.* Weeds will still sprout in the mulch, though their roots won't go as deep, but with just 15 minutes a week of weeding you should be able to keep them under control.* Weeding after a rain makes it even easier.* If you still feel you will use the landscape fabric, plants that grow in clumps will be easier to maintain and do better then plants that spread by surface or underground roots or rhizomes.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.