"She's been opening my letters. That takes the cake!"
Is it similar in meaning to "that sucks"?
"You can't hold two passports. Dual citizenship is forbidden in our country. You can't have your cake and eat it too." OK?
In AmE, "takes the cake" refers to something that is a superlative example of something that is bad or unbelievable.
So, in your example - "She's been opening my letters" - you would need a bit more exposition in order to properly describe the situation as "taking the cake." For example:
"She'd unjustly accused me of having an affair, and I denied it. I explained to her that all those nights I'd been working late at the office I was meeting with my superiors on an important account. She told me that she believed me, but now I find out that she's been opening my letters, and that really takes the cake!"