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If "broke at the base" means it is completely severed, then all you can do is to remove the broken portion, water, fertilize and wait for growth response in spring.

If the basal stem is not completely severed, then prune to remove any overburdened weight that might further the breakage.

Yes you went a little overboard with extreme pruning.

As long as there is adequate soil moisture it will probably refoliate. But the problem is that the response may be rampant growth to compensate for the loss of foliage.

Bouganvillea is difficult to manage as a bonsai, because of the vigorous and bulky growth habit. It's best to do light thinning and shaping regularly, maybe monthly, instead of waiting until it becomes over-grown and you find that you have to do a drastic reduction like that.

Do you prune the roots? Look in a good bonsai book for directions on light root pruning, which may help to slow down the vigorous foliar growth.

Don't over-water or over-fertilize, as that can stimulate excessive growth.

Looks like it is the work of Leaf Miners though you could have multiple pests. Curled leaves and sticky substance are signs of pests and the sticky substance is called Honeydew.
Treat the vine with Neem Oil.
These links have more information.
https://www.gardeningknowhow.com/ornamental/vines/bougainvillea/bougainvillea-loopers.htm
https://www.gardeningknowhow.com/plant-problems/pests/insects/leaf-miner-control.htm
https://www.gardeningknowhow.com/garden-how-to/info/what-is-honeydew.htm
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm

You would likely see damage on leaves of the vine if you had a issue with the Looper Caterpillar.
Chewing of branches could be done by numerous critters; squirrels, chipmunks, rabbits etc.
I would suggest making a small fence around the vine with some hardware wire. This may be the easiest solution without know the culprit in the garden.

https://www.gardeningknowhow.com/plant-problems/pests/animals/get-rid-squirrels.htm
https://www.gardeningknowhow.com/ornamental/vines/bougainvillea/bougainvillea-loopers.htm

Use complete, balanced fertilizers with a nitrogen, phosphorus and potassium (N-P-K) ratio such as 5-5-5 or 10-10-10 to promote good bougainvillea health and plentiful flowers. Too much nitrogen promotes vegetative growth at the expense of blooms, so slow-release fertilizers that moderate nitrogen release are best.

https://www.gardeningknowhow.com/ornamental/vines/bougainvillea/bougainvillea-garden-care.htm

Several insect pests attack bougainvillea leaves, which results in leaf curling and wilting. The bougainvillea looper is a nocturnal caterpillar with a green or brown body. It consumes the edges of leaves, causing them to gradually curl inwards as the damaged tissue rots away. Aphids are also associated with leaf curling in bougainvillea.
Low temperatures also can cause leaf changes.
Leaf curling is a common symptom of many plant diseases, including infections of both fungal and bacterial pathogens.
Initial symptoms include discolored patches on the foliage's surface, followed by withering and disfigurement.
he soil's composition and texture can also lead to curling and discolored leaves in bougainvillea. Heavy soil types, like those with high clay content, often hold too much water near the roots of plants.
I'm not able to tell much from your image, you will need to do closer inspection of the plant itself. These links will help you.
https://www.gardeningknowhow.com/ornamental/vines/bougainvillea/bougainvillea-garden-care.htm
https://www.gardeningknowhow.com/ornamental/vines/bougainvillea/bougainvillea-not-blooming.htm
https://www.gardeningknowhow.com/ornamental/vines/bougainvillea/bougainvillea-loopers.htm