How do I connect a Wii Balance Board to Android via Bluetooth?
TL;DR
Currently it's impossible for Nexus 4 running on stock (AOSP) ROM to connect to Wii Balance Board via Bluetooth.
Reason: On JellyBean 4.2, Android changed its Bluetooth stack, and the new one doesn't &quot;support&quot; L2CAP which is used to connect to Wii peripherals.
Solution: Some custom ROMs (e.g. CyanogenMod) and custom kernels might fix this issue.
(Specific answer for your case: unfortunately, it's not a good idea to make a Unity game for Android using the Wii Balance Board as a controller)

Details
From FitScales's &quot;what's new&quot; and WiiScale's description:


Prevent installation on JellyBean MR1 4.2 due to Google replacing the Android Bluetooth stack. The new stack does not expose the functionality required by FitScales rendering it incompatible for now.

**DOES NOT WORK on NEXUS devices or JELLY BEAN or ANDROID 4.2 devices.



Using that clue as the starting point, I found a discussion on MoDaCo forum:

Post #4: Google broke/removed insecure L2CAP socket support in Android 4.2.x. Wiimotes and other Wii peripherals stopped working because of this. Tell Google to fix/restore this functionality.

And finally, the report on AOSP issue tracker confirms this: Issue #58164: Support Bluetooth L2CAP (consider starring that to follow the progress. Also, there are some constructive technical information posted there).
One thing that might interest you:

Post #50: Alright, so I'm on a Nexus 7 rooted and running the latest official build of CyanogenMod 11. Before I flashed CM, it always asked for a password when trying to connect a WiiMote via straight Bluetooth. Now that I'm running CM, it connects to on without asking for a PIN.

Also, post #113 mentions a link to Android code review which seems trying to support WiiMote (and possibly other Wii peripherals) on Android's next build. However, it still needs kernel that supports the following:

CONFIG_HIDRAW=y
CONFIG_UHID=y
CONFIG_HID_WIIMOTE=y

3.4 based kernels support only the original WiiMote,
3.10 kernels should have native support for 2nd gen WiiMotes,
Wii U remotes did not make it in until 3.11 kernel release.