Howdy rdunlop,
Reading your post, my understanding is that the behavior you're seeing is that the iPhone 6 and wireless speaker are playing unexpectedly in the middle of the night. I don't blame you for seeking an answer on this!
Automatically playing music can be expected behavior when a speaker connects to an iOS device, in some cases. In this specific case, it sounds like the speaker and your iPhone may be connecting during the night. Is it possible that the speaker has a function to "sleep" that is expiring? Another possibility is that the speaker is continuously in a "pairing" or searching mode, and is attempting continuously to connect to the iPhone throughout the night. It'd be best to check the speaker's manual for all of the details, but it might be a good idea to turn the speaker fully off when not in use.
If the iPhone hasn't been restarted in a while, that's a great place to start. Here's how: Restart your iPhone, iPad, or iPod touch
How to restart
- Press and hold the Sleep/Wake button until the red slider appears.
- Drag the slider to turn your device completely off.
- After the device turns off, press and hold the Sleep/Wake button again until you see the Apple logo.

Your step of disconnecting the speaker is a great one, and may help if the speaker is always in a "pairing" or "searching" mode attempting to connect to the iPhone. Fully removing the device and adding it back may help to resolve unexpected pairing behaviors. To confirm, are these the steps you followed? If not, try to remove the device in this way:
Unpair a Bluetooth accessory
To unpair a Bluetooth accessory, tap Settings > Bluetooth, find the device you want to unpair, and tap
, then Forget this Device.
When you Forget this Device on your iPhone, iPad, or iPod touch, that accessory is removed from the list of available Bluetooth devices. To add the Bluetooth accessory to your device again, place it back in discovery mode and repeat the steps above to pair it again.
The full article is here: Set up and use third-party Bluetooth accessories
Once removed, you'll be able to pair the speaker again.
Cheers!