Disconnected Wires; Soldering Necessary?

Hello! I've taken my old Tamagotchi apart for repairs, but it seems one of the little wires popped off the... speaker cell? The little round part that is not the battery. Do I need to solder it back on? If yes, what soldering kit should I get? I know I would need something with an extremely small tip, but I have never soldered anything before. Is there a certain temperature to use? Anything else I need to know about soldering small things?

For clarification's sake, one of the white wires in this picture is what has detached.

Thank you for reading!