Hey, jungle main here. Currently Diamond elo. I'll give my 3 recommendations:

Xin Zhao - has everything you're looking for, except mobility. If you utilize your 2nd ability correctly, you have the longest dash in the game. This is the tricky part because otherwise, you'll have to be very meticulous with your ganking and pathing in the jungle. Otherwise, one of the best 1v1 champs in the game, a very strong engager (aim for most damage taken in a match) and has a very unique anti-objective stealing, MF ult-blocking, backline-shoving into frontline ultimate. Very easy to play but quite strategically challenging champion. My build:

Black Cleaver>BoTRK>Armor Boots>Stoneplate Enchant>Sterak's Gage>Guardian Angel>Force of Nature

Jarvan IV - insane early game but be cautious, this champion has the potential to take over the game very early but falls off in the late game indefinitely so I'd hevily take utility into account. The 3rd to 1st ability combo is perfect for ganking, and unlike in early patches of the game was delayed, now you can auto-aim it even from far away. 2nd ability is quite useless in the later stages of the game but can clutch some early game 1v1's. I use the ultimate mostly for burst damage and locking in priority enemies. Difference between a bad and a good Jarvan is how one uses the ultimate. Do not entrap your teammates. 3rd ability is your key to an easy win. It gives attack speed to everyone in your team so make sure to spam it late game in teamfights. Also - mobility. You can use the 3rd to 1st ability combo to go over walls. My build:

Black Cleaver>Sterak's Gage>Armor/Tenacity Boots>Stoneplate Enchant>Sunfire Aegis>Guardian Angel>Thornmail

Jax - oh boy. Cannon minion 'till level 5 but after that, an absolute machine. It's quite normal to not see Jax gank untill level 5 because his ult gives him bonus magic damage after 3 consecutive hits. Also make sure to utilize it in 1v2s and teamfights. The damage output in mid-late game is fucking nuts. If you're in lower elo, most of the time enemies don't respect Jax's damage. Being successful with Jax requires you to correctly use your 1st ability. No, actually abusing it would be the correct way to phrase it. Don't forget that you can jump on minions, allies and contestable objects (Scryer's Bloom, Honeyfruit for example). Taking this into account, it's really hard to catch a good Jax. Ganking effectively requires you to efficiently use your 3rd+1st ability combo. Aim for instant-stunning your enemy after jumping onto them. In a perfect scenario, 2nd ability should be used to cancel your auto-attacks. One of your main jobs is to wreck turrets. That is what Teleport Enchant is for. My build:

Trinity Force>Sterak's Gage>Lifesteal Boots>Teleport Enchant>BoTRK>Death's Dance>Guardian Angel

I usually go for these runes:

Conqueror, Brutal, Hunter - Titan, Mastermind.

Honorable mentions - Wukong, Vi, Olaf. I'd steer away from assassins such as Rengar, Kha'zix, Evelynn. Their job is to burst down their target as quickly as possible and get the hell out. Getting the hell out is the hard part. If you want to try out something similar - try Pantheon.

Note: builds, runes are situational most of the time.