The EXACT same thing has happened to my daughter's phone 2 nights ago.  NOTHING will charge her phone anymore -- and it all happened within 2 days of upgrading to Lollipop. I honestly think a factory reset will fix it, but I can no longer get any charge to the phone to power it on to reset it.   I have ordered a new battery, cradle charger, charger, and cable (wanted to cover all bases) from Amazon and everything will be here later today.  I'll let you know if anything helps.  As reluctant as I am to factory reset her phone (she has more crap on it than any normal person should), I think it may fix the problem, but I need to get a charged battery first.