As WineSoaked says there are dozens of possible solutions.
I'll suggest one.
You can connect 8 MCP23017 port expanders to the Pi's I2C bus. This will give you 128 inputs and outputs.
You can use 60 to drive the LEDs and 60 to connect to switches, leaving 8 spare.
On the output side you would just send an I2C message to switch a LED on. On the input side you could either poll for switch changes (i.e. regularly read the chips, perhaps twice a second) or connect the MCP23017 interrupt lines to a gpio (so you would only need to poll when you get an interrupt).
This solution wouldn't scale very well if you needed twice as many LEDs, but you could always buy another Pi.