This does look like a sedum. I'm not good at specific identifications. I leave that up to the experts, like @stormy, who has already given you that information. However, because I have a lot of sedum plants, I can tell you they're extremely easy to care for and will indeed flower.
I have two types, one is a "creeper" which has a trailing habit, and one has an upright growth habit. The growth and care instructions, at least in my case, are the same.
Sedum is as close to a "set it and forget it" plant as you'll find. You're already experiencing that as it came back really quickly after having a problem. It thrives in most soil conditions, and all levels of sunlight, including shade. The one I have in the shade (not pictured here) doesn't grow as quickly or flower as well as those in full sun, but I don't mind because the foliage stays pretty and it adds interest to an area that other plants don't like.
Sedum grows in a wide range of climate zones, from 3 all the way through 9, depending on the variety. I live in zone 6.
The World of Succulents site is very informative. This page is dedicated to the Hylotelephium spectabile (Sedum spectabile), and has a lovely picture of one in bloom.
According to them:
They are ideal for that part of your yard that gets too much sun or too little water to grow anything else. A common name for Sedum is Stonecrop, due to the fact that many gardeners joke that only stones need less care and live longer.
It looks like yours is big enough to separate into smaller sections if you have different spots to fill. Some people say it's healthier to split them every few years. I don't bother, and only split them when I'm giving a piece to a friend. They make a great gift because the receiver doesn't have to do much work, as opposed to other plants, which get fussy when moved around.
Transplanting sedum is very easy. See my answer here for general transplanting instructions which I've used successfully. (That answer also has pictures of sedum pieces which I thought had died, but were perfectly healthy once I planted them.)
However, if you don't feel like going to very much trouble when transplanting, you can even just put them on the ground where you want them to take root.
For shorter varieties, simply laying the Sedum on the ground where you want it to grow is normally enough to get the Sedum plant started there. They will send out roots from wherever the stem is touching the ground and root itself. Source.
The pictures below are my plants as they look right now, in mid-August. As you can see, these are getting ready to bloom. They start in mid to late August, depending on where you live, and last deep into the fall, sometimes until the first frost. Some people choose them specifically because they provide some color after other perennials have gone by.
The top left picture is my creeping plant. It has small clumps of flowers which will be dark pink/light purple. They're sort of sparse looking though. I prefer the shape of the upright variety flower, in the top picture on the right. As you can see, it will be more full. Mine is a bright pink, although they come in many colors.
It's now September, and my sedums are blooming profusely. In fact, they're just on the verge of going by. I've added pictures of both the creeping, and the upright, varieties. On the top row are the pictures from when I first posted this answer in August. Below them are the two varieties taken today. If you look closely at the flowers on the upright plant, you'll see a large bumble bee. They absolutely love this plant!
Click on the pictures for a closer, better, view.