Cannot repro.
Please provide the device info, including Android and Chrome versions.
Also could you please open DevTools on DevTools (undock DevTools and press F12)  and check if there are any error messages in the Console.

So the android version didn't change since it was working before, but is the following:
Android version 5.1.1
Chrome stable on device (for some reason this fails to update every time even after reflashing the Nexus 4) 40.0.2214.89 (Official Build)

No errors present in dev tools on dev tools.

Oh it's also worth noting I get the same issue trying to connect to my Pixel phone:
Android 7.1.2
Chrome 59.0.3071.125

I see a similar issue with:

Google Chrome: 59.0.3071.91 (Official Build) (64-bit)
Revision: 0
Platform: 9460.60.1 (Official Build) stable-channel chell

Phone is a OnePlus 3T, Android 7.1.1, Oxygen OS 4.1.6.

1. Visit chrome://inspect/#devices on my Chromebook, nothing happens.
2. Untick and then re-tick the 'Discover USB devices' checkbox.
3. I get an 'Allow USB Debugging' prompt on the phone, click 'Allow'
4. Back in chrome://inspect, no devices are shown.

The same behavior exhibits when I use the remote devices via devtools. I have to untick/re-tick 'Discover USB Devices', but even after allowing that nothing happens.

I do not appear to get a pending authorization UI, so might not be exactly the same issue.

FWIW, this works for me in 58.0.3029.140 (Official Build) (64-bit) (the current recovery image) but then stops working after the auto update to 59.

Re #1, did you try on ChromeOS? I suspect that it is specific to the adb support there. I'm unable to do any remote debugging as a result - perhaps this is a P1?

I've bisected it down to having stopped working sometime between ChromeOS build 9334.72.0 and 9460.64.0. I'm currently having issues getting simplechrome to build chrome to see if it is a chrome regression and bisect further.

Resolved?