Those inductors are soldered to the board like any other component on the board - that is metal in the component is soldered to metal on the board. That sort of component is a little tricky to solder by hand as the majority of the contacts are underneath the body of the component. When they're initially installed they're placed on top of a layer of solder paste on top of the contacts on the board, then the whole thing is stuck in a reflow oven to cook it and cause the solder paste to melt to liquid solder.
To do them by hand I'd melt a bit of solder onto the board contact then squish the component down onto it. Both sides need to be connected for it to work, like any sort of two-leaded component.
I'm slightly surprised anything works with that disconnected, as I think it's was part of the voltage regulation circuitry, although I suppose an inductor there is probably just suppressing high frequency noise, which should be pretty minimal when running from battery anyway.
Regarding the hole in your lid, there is nothing there on the right side iirc. If you'd gone into the left side you'd have hit the wifi board, but I think you've missed the speaker on the right so you're okay. It's up to you whether you're happy with the look of it of course.