Going back a bit, when you removed the battery did you pull it and put it back in right away? Try removing the battery, leave it out for a couple of minutes, and then put it back in to see if your Note will start up again. (Leaving the battery out for a few minutes will allow any trace power left inside internal components like capacitors to fully drain.)
If that doesn't make any difference, keep trying to get into Recovery Mode. Power off your Note and when you restart it, hold down the Volume Up button, the Home button, and then the Power button simultaneously. When the screen shows that ‘GALAXY Note II’ you can release the Power button but continue pressing the Volume Up and the Home buttons until your Note finishes booting up into its Recovery Mode. (Just for future use, keep in mind the Volume Up button is to get into Recovery Mode where you can do various low-level maintenance and system tasks. The Volume Down button is to get into Download Mode which is for doing things like installing a third-part Recovery or ROM if you want to root your phone.) Anyway, as Bg260 already mentioned, select the 'wipe cache partition' option and clear the system cache. That will only take a few seconds and then reboot your Note to see if it will start up normally again.
If that doesn't work, you might want to think about re-flashing the factory ROM. Re-flashing the ROM is basically reloading the Android operating system. Directions for the process:
http://updato.com/how-to/how-to-install-an-official-samsung-stock-firmware-using-odin
This is based on using Odin, a Windows based utility. You can find the latest version of Odin here:
http://odindownload.com/
But the latest version might not be compatible with your Note II so maybe try v1.85:
http://updato.com/how-to/what-odin-version-you-need-for-your-phone
Also, you might need to install a Windows driver so it can mount your Note properly:
http://www.samsung.com/us/support/downloads
If you don't have access to a Windows box, there's also Heimdall, the Linux and OS X (macOS) equivalent of Odin:
http://glassechidna.com.au/heimdall/
And the big download will be the firmware for your Note. It's very, very important to use the exact firmware for your specific model and carrier:
http://samsung-updates.com/device/?id=SGH-T889
Flash the ROM is something of a daunting project but if successful it will only replace the ROM. Your personal data is stored in a completely different partition than the system partitions so your personal stuff should still be left as is. However, if that doesn't work you should try doing a Factory Reset, but don't forget this will wipe clean the general user partition, where your data is stored. Doing a Factory Reset is more of a last resort step. In your case since your Note isn't starting up normally, hopefully you have anything important already backed up.
