Carb sync is not his problem, if he is going to sync them, bench sync is the only proper way to ever sync them.
From the video it's a NewGen model.
Sounds like the pilot circuit is clogged. I'm also guessing that there's no fuel filter either.
Personally I'm thinking improper use of the choke still, after watching the video several times, I noticed that the choke was already on during the first attempt, which seemed a little short time of a try, then he switches the choke off, and tries again, and seems too short again of time to crank it over.
To use the choke: On the 250, you pull the lever toward the rider to apply choke, then push it away to disengage. Turn ignition key to on position; leave choke off. Press starter button with one finger (no throttle) while gradually turning up the choke. After about three seconds, the choke and carb priming should be adequate to start the engine. Use no more choke than necessary to keep the bike idling. Idle speed at startup will most likely need to be slightly higher than when it's warmed up.
The gradual application method is better than just turning on the choke and then trying to start it, because the Ninja rarely needs full choke to get going. You don't want to give it too much choke, because you risk flooding the engine.
As soon as the engine runs without dying (from several seconds to no more than 60), ride off with the choke still on. Do not idle the engine to warm it up. Keep the revs low, and gradually lower and then turn off the choke when the bike can run fine without it. This may be a few blocks or a few miles. Increase revs as the bike warms up, and don't run it hard until it's fully warm. Make sure to check a couple minutes into the ride to make sure you're not riding around with the choke left on.
On the other hand
Some people have trouble with the gradual-on method. Here's a way you can try if you're not having much luck: Set the choke lever at about halfway, then hit the starter. Normally it will fire right up and the rpm will quickly jump up. Turn the choke up or down to keep the bike running, staying under 3000 rpm until you start moving. Ride off easily, closing the choke completely within a few blocks.
Remember that every bike is different, and you need to find the method that works best for you.
Also once you touch the throttle, the choke is useless due to the design, it's an enrichment circuit, once the throttle butterflies are moved the circuit is rendered dead.
One must remember that it's NOT FUEL INJECTION, it's old school carburetors.
Or is there something he's not tells us? We need all the details to offer you the best advice, leaving out details will not help anyone.