You need to read data from the GPS and parse it with the TinyGPS(Plus) class. The TinyGPS(Plus) class does not read data from the GPS by itself. Here is a guide to look at.
EDIT: If you look at the code you posted, the "if" statement before the one that checks for button press prints latitude and longitude to serial monitor. Do not use the "String" class for anything on AVR based boards, there is always a better way of storing stuff, two floats would probably be ideal.