An outline of one possibility:
Debounce your switch.
When the switch becomes pressed (IDE/file/examples/State Change Detection) toggle a boolean variable. Call it buttonWasPressed.
Use SCD again to detect when buttonWasPressed goes false-to-true and also when it goes true-to-false.
Use the t-f edge of buttonWasPressed to trigger a timer for LED1. Use the f-t edge to trigger a timer for LED2. Or vicey versy.