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It most likely isn't getting enough light. That is the biggest reason plants don't bloom indoors. Begonias need bright but indirect light such as from a south facing window. They also need humidity such as from a pebble tray filled with water. A fluorescent or grow light can help if window light is not available.

The next thing could be lack of nutrients. How long has it been in a container? It may need the soil replenished or a slightly larger container. Here are their growing needs you can compare with yours:

https://www.gardeningknowhow.com/ornamental/flowers/begonia/begonias.htm

Have you fertilized it at all? You don't say what type of begonia you have, but generally they can do with light fertilizing. Here is more:

https://www.gardeningknowhow.com/ornamental/flowers/begonia/tuberous-begonia-fertilizing.htm

Yellowing leaves can indicate improper watering, so check that first. Begonias are easy to overwater. Do not let the leaves remain wet. Rex also needs high humidity, so you might consider a pebble tray. Since those look like raised bumps in the photo, another possibility is edema.

Here is more:

https://www.gardeningknowhow.com/plant-problems/environmental/treating-edema-in-plants.htm

https://www.gardeningknowhow.com/ornamental/flowers/begonia/growing-rex-begonias-indoors.htm

Unfortunately, your photos did not come through. I am unable to see the damage. It could be galls caused by insects, or fungal/bacterial infection.

https://www.gardeningknowhow.com/plant-problems/environmental/treating-leaf-gall-on-plants.htm

Don't be afraid to trim these tuber shoots back. If they are given proper soil and growing conditions they should come right back, hopefully strong and healthier. Definitely, get rid of those leggy ones. You can leave some of the stem with a node above the soil line, but even cutting them back to the tuber shouldn't be a problem if you need to.

Here's some helpful info:
https://www.gardeningknowhow.com/ornamental/flowers/begonia/growing-begonias-from-corms.htm

It's fine - roots can sprout from all over the tuber. You can bury it or let it stick up out of the soil. Depends on how developed it is. If it's long enough to have formed a leaf, you may want to let that be above the soil line if you can still bury the tuber into around an inch of soil. Otherwise, it should be fine to cover it over. Here's a helpful article:
https://www.gardeningknowhow.com/ornamental/flowers/begonia/growing-begonias-from-corms.htm