Can you break in with the keyboard attached? If so, what does PRINT PEEK(64447) give you?

• if PRINT PEEK(64447) returns 255, you have an original IIc (ROM revision '255').
  
• if PRINT PEEK(64447) returns 0, you have an Apple IIc with UniDisk 3.5 inch drive support (ROM revision '0').
  
• if PRINT PEEK(64447) returns 3, you have an Apple IIc with memory expansion support (ROM revision '3').
  
• if PRINT PEEK(64447) returns 4, you have an Apple IIc with a modified/fixed version of memory expansion support (ROM revision '4').

You can order ROM upgrades from ReactiveMicro:

http://www.reactivemicro.com/product_info.php?cPath=1_28&products_id=53

But alas, "dealer upgrades" are no longer free. ;-)

It sounds like your 5.25" floppy disk is the next thing to worry about. It can also be taken apart and treated to TLC: reseating of chips, etc.