Answer to "How do you fix the Blue Inductor Coil on iPhone 4?"I just dabbed a small blob of conductive silver with a circuit pen across where the coil was and it works perfect now.
Answer to "No Service after Teardown :/"it is more than likely the little blue antenna coil (no more than 1mm in size) highlighted in the photo in the previous answer. this happened to an idiot "friend of mine" who changed his screen with the instructions on this site (ok it was me). perfect instructions to change the screen from this site but didn't see any advice about just how fragile that coil is. so after pulling apart my phone and reassembling it four times (was jipped into just buying the digitiser screen before realising you need that and the lcd as the lcd doesn't detach, then getting a faulty part) i knocked the coil off. if you aren't electronics savvy and aren't confident of putting in a new one yourself you can do what i did. i used a circuit pen and put a tiny dab of conductive silver bridging the connecting points where the coil are and now the phone has perfect reception again (actually seems stronger than before). you can get one of these pens from any hobby electronics store. good luck -dan