How do I turn this into a flower bed?

What happened to it? It looks like it gets a lot of shade but also looks like it was burned or dug out. Anyway, you could put a boarder around it with stone or wood or block and build it up for a raised bed. As it looks like it is in shade you will need to choose plants that require shade. Hostas, coleus, bleeding hearts, ferns and such. I like a raised bed as it keeps the mulch in as well as adding interest.

Sorry to give you bad news but if your area won't grow grass, it probably won't grow anything else. Looks to me, you have some sort of chemical or diesel fuel spilled in your yard. Smell the soil to see if you can detect the culprit that's causing the problem. You will most likely have to replace 10 inches of the soil with new soil. Then you can seed it down to whatever you like.

As long as it is smooth, it won't matter whether you use wood, block, stone, or rubber edging. You will only be able to get close and then you will need to weed eat close to it. I would alternate between mondo grass and hostas around the edges for a living border. Then fill in with plants that do well in the shade like coral bells for color and height, dead nettle for color and ground cover, astilbe, Japanese Forest grass in a couple spots for a "mound of green", browallia hybrid is extremely low maintenance with color and mounds, garden hydrangea that is a shade version, geranium cranesbill for a bit of color that is low to the ground and blooms all season. All these suggestions are perennials.

Do you have pine trees around that area? If you do, there aren't many things that will grow under the canopies. If not, have you tried digging down to see if perhaps the area has rock or something underneath the surface? You could also have the soil analyzed to see if perhaps that was the problem. When we lived in Texas, the home we purchased was built on an old hobby farm and the gravel driveway ran through the back yard. The didn't take out the gravel, just sodded over it and we ended up with a strip of grass that burned out every summer from the heat on the rocks under the soil. Watch how much sun each section gets and plant according to that, it is a pretty big area and you could have three or four different sun type areas where you will be able to plant different sun requirements. Good luck in finding a solution to be able to plant your garden.

When it comes to beds, there is no such thing as "easy to mow around." You can try banding your bed with metal edging that is hammered into and then set with spikes in the ground to see if that stops lawn creep into your bed, but you will still need to trim with a string trimmer or electric scissors (I have electric scissors/clippers on a long extension handle that has wheels on the bottom that works beautifully and spares my back and aging knees; takes longer than a string trimmer but also doesn't tangle; battery operated and recharges quickly). Line the inside of the metal edging with rocks, bricks, or concrete edger blocks, whatever look you like. Pinterest has tons of ideas.

I'm wondering if that soil has some kind of fungus in it that would kill the plants like it killed the grass. Has it spread year after year, starting out as only a small area that you didn't pay much attention to at first? Maybe you need to dig out the "contaminated" area and replace with new top soil and compost before you try planting anything. It would be awful if you spent money and time planting a flower bed only to see it fail because there's something in the soil that is killing the plants. Another option would be to line that entire area/cover with landscaping fabric and build a raised bed on top of it, as Darlene suggested.

I'd suggest sticking a couple of shade-loving plants in that area to see if they will grow. Did you already clean out that area and is that why it looks as it does?

IF you find the plants will grow, use any edging you can afford to create the border per suggestions above.

If the plants won't grow, I'd suggest filling that area with mulch or other things such as decorative gravel and grow your plants in pots.

Okay, so you said, "flower bed or something nice". The something nice will make you a LOT happier! First, cover the bare area with a weed block fabric product. Then, you can edge it with metal edging, those made-to-look-like-rock concrete stones or whatever you like. Once the liner and edge are in, fill it with a mulch. It could be cypress or pine, rubber or even stone. You now have a backdrop for whatever you think "something nice" might be. Options: a tall rock garden/fairy garden, a fountain, bird feeding stations, a raised bed of flowers or some nicely arranged potted plants (that love shade), even some lawn art. Or, you could do any combination of the above, depending on your whimsy and budget. Most of all, enjoy being outside and creating something beautiful.

Do a hosta and fern garden. You can easily mow around hostas. There are a lot of varieties to choose from. Pairing these two together can have quite a dramatic look and wouldn't require a formal layout. In other words, easy to do.

You just need some shade loving plants. Hostas, impatients, and other shade plants would look amazing.