See if the following code will do the trick for you:-
Application.ScreenUpdating = False
Application.DisplayAlerts = False
Range("L1", Range("L" & Rows.Count).End(xlUp)).AutoFilter 1, ""
Range("A2", Range("K" & Rows.Count).End(xlUp)).Copy Sheet2.Range("A" & Rows.Count).End(3)(2)
Range("A2", Range("K" & Rows.Count).End(xlUp)).Delete
Application.DisplayAlerts = True
Application.ScreenUpdating = True
The code will filter on Column L for blank cells and will transfer the relevant row of data to sheet2. The code will also delete the "used' data from sheet1.
Following is the link to my test work book for you to peruse:-
Click on the button to see the code at work
I hope that this helps.