In order to enter recovery mode, you will first have to turn off your device. Once it’s turned off, press volume up + power button + home button simultaneously and keep them pressed until you see the recovery mode.
If you haven't already, try booting in recovery mode, you might need to let go of the buttons after a second or two for the boot to continue. (It's been a while since I've had to do it this way, and I can't turn my phone off to try at the moment.) If that works, try a factory reset, i.e. clear user data.
If the above doesn't work, the hardware could be at fault. Am I right to assume you haven't modified the phone in any way?
Depending on what Rogers quotes you for a repair, you could get another Note 2 off classifieds (be careful with provenance as you wouldn't want a blacklisted device) or get a new one from Koodo (a Telus subsidiary) who sells them for 300$, unlock it (free from here, cheap from third party vendors, expensive from the carriers) and be on your way. (If you want your Rogers apps back, there's a code on here to change your Carrier Specific Content code from KDO to RWC, this will clear your data so do it early on.)
One last thing: the Rogers/Bell/Telus Galaxy Note II is the SGH-I317M, which shares the AT&T Note II forum, not this one.
My phones: Galaxy Note II SGH-T889V (with LTE enabled
/ ext4 SD card support on TouchWiz ROMs
), Motorola XT-1045 Moto G LTE (TWRP
Previous phones: Nexus S GT-I9020A, Galaxy Gio GT-S5660M (Arpeggiomod ROM
/ Discussion, stock rooted ROM
), Motorola XT-1032 Moto G.
Phones for which I play "tech support" for family and friends: Galaxy Nexus GT-I9250, Galaxy Q SGH-T589R, Galaxy 551 GT-I5510M, LG C-800G Eclypse, Motorola XT-1032 Moto G