NOTE: All of this is for boxing only. MMA changes everything.
Get your left up AND KEEP IT UP!!! That left has no business being so low.
If you have the speed of Camacho, Leonard, or Ali you can get away with holding your lead low. Us mortals need to keep it up. Of course, you can always drop it to draw an opponent which all three of those greats often did.
If you watch fights by those three you will notice that their opponents (typically the slower fighters) usually kept their lead up. They had to or they would eat fast jabs all day long. Only the very fast fighter could get away with holding the lead low.
Leaving that left low:
1. Exposes your head. Did you notice that your're taking a lot of shots to the left side of your head?
2. To avoid being hit you are bending at the waist and moving your entire torso and head. This is a wast of time and energy. If your sparring partner wanted to he could rush you at that time (which he did, just not as much as he could have) and land multiple shots while you struggle against your rearward momentum.
3. Bending so severely at the torso is eliminating your ability to counter. At the least it increases the time required to counter which effectively reduces your speed.
4. That low hand his killing the speed of your jab because it greatly increases the travel distance. It is also easier to perceive because it is taking an angular path (which is easier for binocular vision to perceive) to its target.
5. That low hand is also killing the body mechanics of your jab. While it is not a power punch the shoulder does play a role at the end of the punch. Delivering it from that low position makes it an arm only punch.
If you;'re concerned about body shots, use your elbows and forearms to protect that area. This is extremely fast and leaves your gloved hand in position to protect your head. This also lets you stand more erect. Both of you are bending awkwardly at the waist.
You've also got a lot of weight on your back foot. That eats a lot of speed because you have to transition much more weight to effect a forward action. Watch some Ali film and stare at his feet. Look closely at your film and notice how much time you are being driven backward. Get some of that weight off of that back foot and this won't be happening.
Try delivering that jab as a straight punch from a position similar to how you are holding your right. It will definitely score more often. It will be much faster and, when you want to, you can add a little shoulder behind it. You will cut the travel distance in half, it won't be nearly as telegraphic, and you will be in position to counter behind it.
The only times that lead should be held low:
1. When you are the faster fighter and can get away with it.
2. When you have a distance advantage and can get away with it.
3. When drawing your opponent.
4. When you put yourself in position to rest your lead.
Lastly, rechamber all of your punches in the most direct path possible.
Hope that helps.
Edit: When you combine your faster jab with better weight distribution, and less concern for protecting the left side of your head, you're going to be driving your sparring partner back. He's going to be fending off your jab and you are going to be in position to take advantage of it. At the very least you're going to meet him in the middle instead of being driven backward.