Maybe she didn't get the memo that said that she had to go there!!
2' x 4' seems kinda small to me, but, I've been wrong before!
My guess is that you don't have a fenced yard, right? How about buying an x-pen and setting it up in her designated area? As far as "ruining the lawn", just hose it off after she goes on it. Yorkies don't pee THAT much!! When Ginger was young, I had to take her out on a leash. She woldn't go unless I had my back turned to her.
Everybody wants a little privacy sometimes!!
Suz