To me, leap, bound and spring have slightly different emphasis compared with the more neutral noun jump. At least to my ears:
a leap emphasises the distance, or gymnastic ability of the jump.
a bound emphasises the energy or perhaps the enthusiasm of the jump.
a spring emphasises the speed or the unexpectedness of the jump.
So for example, I might use them in the following sentences:
The cat leapt into the air, and landed on the next door neighbour's balcony.
When Joe came home, he was met with Bruno the dog, who bounded through the hallway and jumped up to greet him.
When Mary stepped on the cat's tail, he sprang up and scratched her.
In the context of your sentence, where the emphasis is on the nimbleness and perhaps distance of the jump, I would therefore choose leap, although I would avoid mixing leap with sprung, instead preferring a more neutral noun to go with the verb leapt or a more neutral verb to go with the noun leap:
In one leap, she jumped up on her desk, nimble like a cat.
In one quick movement, she leapt up on her desk, nimble like a cat.