It looks like you have extremely water retentive soil. Depending on how badly rotted the plant it's possible it can't be saved; What i'd recommend you try to do is swap out the dirt with cactus mix with 50/50 perlite thrown in. It'll keep water a LOT less than the soil you have right now and it'll help prevent your plants from rotting away.

Doesn't look like String of Pearls, but regardless sorry to say it's dead:(.

@kara I LOVE the way the Mammillaria Hernandezii looks in the white pumice. It's a very nice aesthetic. I am using 100% pumice as my rooting medium for cactus. I think the very light nature of the soil is very helpful to the new root system and it is able to expand out widely and develop roots rapidly without expending energy moving heavy soil particles. So far I have not used pumice as a soil ingredient. I will probably start using horticultural lava soon, which aesthetically I far prefer to pumice. I am also thinking of buying an off-the-shelf soil product made by Gary at Laguna Beach Nursery named Top Pot. Gary limits organic components in his soil mix to 30%, which stops the collapse of the soil and allows it to live for five years or more without a change. He uses peat for that 30%. He uses as inorganic components pumice, sand, and perlite. I find that an odd mix. I am having problems finding a local supplier. I could create my own mix based on his idea, and as a worst case I will do that. But the time to create soil mixes has value to me as well. My suspicion is that String of Pearls would thrive in such a soil. I fully agree with the idea that succulents can be raised in 100% small-particle organic soil and still do well. You simply need to water them with extreme care and make sure the environmental exposure allows for rapid evaporation of the soaked soil. But this requires an expert gardener who is dedicated to plants. What I thought I was getting with a structured soil was the ability to be a lazy gardener who could just abuse and over-water my succulents and still have them thrive. It turns out that it is not that simple.

String-of-pearls likes bright light, but not direct sun. The LED grow light will help. As LED lights are cool to the touch, you can set the light very close to the plant without burning it. Note that as your plant grows you will need light that reaches the leaves hanging over the side of the pot. Hang the plant in a bright window when the strings get longer. Your plant looks good. The pearls are fat. The new stems are longer than they should be, but that is probably because the plant wasn't getting enough light. The LED light should make growth be more compact. String-of-pearls grows fairly quickly. It is best to let the soil dry out between watering. Some people only water once a month, but you should keep an eye on the leaves to make sure you are watering properly. If the leaves look wrinkled the plant probably needs to be watered. Essentially, when dry the interior of the pearls (the leaves) shrink and the outside shrivels slightly. Be cautious, though. Too much water can cause root rot that can also make the pearls shrivel because the damaged roots can't take up enough water. It's best to soak the soil and allow the pot to drain well before you set it back on its saucer. Young plants tend to need more water than older plants, but err on the side of too dry.