can someone tell me whats wrong

[Bryson89]

i got a 1981 ford fairmont and its had some elctrical problems with starting it one day it was kinda cold outside an my car had been sittin a little bit wen i started it it ran a little bit then died then started and it was jus to cold and i didnt plug it in long enough well now ive had it on the batter charger over night tried jumpn it and nothin seems to work when ever i turn the kye it jus goes click an thats it as if my battery is dead but thats not it i looked at my solanoid and there were two pins on the face plate that were poken out i thought maybe thats it but when i tried putten the little pins through the plate it jus goes there an holds it there i think but it jus broke a little peace of plastic off the back of the solanoid so do i need to get a new one or does this even matter its still doin the same thing i donno what it is can n e one help me out