Apple wireless bluetooth keyboard won't pair - have tried everything

I have an Apple wireless bluetooth keyboard that I can't get to pair with my late 2013 macbook pro. I've tried every solution I've found online, but I'm not sure if they apply to my situation.

The keyboard gets discovered and I attempt to pair. The pairing screen comes up, and I type in the passcode and hit return, but it never works. I'm wondering if for some reason the keyboard is not sending out the keystrokes. I've tried to pair it with 3 other computers (two other Mac laptops and a Mac Pro) with the same results.

Here are the things I've tried (which may provide clues)

- I replaced the batteries with completely fresh new ones (non-rechargeable) from the store.

- I tried resetting NVRAM on my laptop.

- I tried resetting the SMC.

- I tried "resetting the keyboard" (i'm not sure what that means technically) by holding down the power button for several seconds, then turning it on, continuing to hold down the power button while I'm bringing up the pairing screen. This didn't seem to do anything.

- I fired up an old laptop which I haven't used in years, which is probably the last thing this was paired to successfully. Interestingly, the laptop said it was paired with the keyboard. But I couldn't get it to respond to the keyboard. It just wasn't receiving the keystrokes. I unpaired it from the old laptop. Tried to pair it again to the old laptop and I couldn't. Same results – I type in the passcode and nothing happens. This is part of what makes me think the keyboard is not sending out keystrokes properly. I tried the other authentication methods that the older "bluetooth setup assistant" lets you try (no passcode, blank passcode, custom passcode, etc) – none of them worked.

- I investigated whether it might be in "mouse keys" mode or something (but I think I understand that this is something that's done by the computer, not the keyboard itself). This doesn't seem to be enabled on my computer. I found something somewhere about pressing the alt key five times in quick succession to get it out of this mode – tried it, didn't work.

- I tried all of these things in different orders.

Clearly the keyboard and the computer are seeing each other and communicating to some degree – it shows up when the keyboard is on, doesn't show up when it's off. Only allows me to try to pair when it's on. But the pairing never happens because my computer never hears the passcode from the keyboard.

I'm out of ideas. Anyone else have any?