First, you can use other ways to address cells other than .range().  In your
case, .cells() looks like it work much nicer.

.cells(x,y).formula

The x represents the row and the y represents the column.  And y can be a number
or a letter--excel will accept either.

I _think_ that this may be closer to what you want:

Option Explicit
Sub testme()

   Dim iRow As Long
   Dim iCol As Long
   Dim Max_Row As Long
   
   'test data
   iRow = 3
   iCol = 3
   Max_Row = 777
   
   ActiveSheet.Cells(iRow, iCol).Formula _
      = "=index(K2:K" & Max_Row & ",match(A" & iRow & ",l2:L" & Max_Row & "))"
   
End Sub

Notice that I changed (max_date) to max_row.  I can't think of where I'd want
the range in =index() to have different number of rows than the range in the
=match() portion.