Is there a way to still charge a nabi 2 without using the little round charging port?

The connector inside the nabi broke loose

Here are trouble shooting tips from the Nabi site:
https://www.nabitablet.com/support2/fb/top-10-faqs/nabi2/2

Also, there appears to be a problem with the wall charger so it's better to charge it via USB to computer instead of from the wall outlet. That might be the reason why your Nabi won't turn on.

Here's more information regarding the charging problem:
https://www.facebook.com/nabitablet/posts/301200709922574

If it truly is the connector in the Nabi, you're in a bit of a jam. But, the power cords I find are very finicky. I have two Nabi's and each has its "preferred" charging cable. One cable works better than the other on each tablet. Before attempting any major repair, try another USB charging cable. You can get the cable for under $4 on Ebay. Make sure you have a high powered USB source (2 Amp or 2000 milliamp) or it won't provide enough juice to charge the tablet. Beyond that, I also sometimes have to "position" the cable just right, twisting the connector and cable till I get the red charge light to stay on. Often this requires I fully power off the Nabi, not just put it into stand-by. Good luck.

Try reseting it by holding down the power button (while unpluged,first) for 5-10 seconds. Then plug in charger.

Sounds like you have a good charger/cable, so try resetting the Nabi that's giving you trouble. Hold the power switch down for 10-20 seconds to reset the unit.
I have a couple of NABI-2 tablets and they both react differently to the charging cables; I often have to shut the NABI off completely to charge them. Also, getting the charging cable to connect can be challenging - I will plug the cable in and out several times waiting 10 seconds or so to see if the green LED will go red.

If the end of the charger broke off inside your nabi, it will be difficult, but you can get it out. Ideally you still have the broken piece. Superglue the end, put it back in and let it set, then pull it out quickly. If you don't have the rest of the broken piece, you can use something like a toothpick or a cotton swab with the cotton tip removed. Check out YouTube for videos on this. If your charging cord just fell apart and nothing is broken inside your nabi, just buy a universal charging cord. Make sure the tip matches the one you broke. Amazon is a good place for finding these. It will still charge your device.

Try a different charger first, as it maybe faulty. If not the battery maybe faulty and I would purchase a new one.

Press "Volume +" together with "Power on", during 8-10 seconds. Choose first line in menu.

I fit is a new tablet, I would take it (/send it ) back to supplier and complain and/or ask for a replacement.
If it is not new, then... hmmm.

Whenever you plug in your Nabi 2 tablet to charge, The tablet will boot up. Whenever tablet boots up, turn it off and leave it charging for a good 4 to 6 hours. That should charge it.