Step 1: Instead of dividing $6!$ by the factorial of the repeat count, count the arrangements directly by first fixing positions for the repeated letter.
Step 2: The word LEADER has 6 letters: L, E, A, D, E, R. The letter E occurs twice; all other letters (L, A, D, R) are distinct.
Step 3: First choose 2 positions out of the 6 available slots to place the two identical E's. This can be done in $\binom{6}{2}=\frac{6\times5}{2\times1}=15$ ways.
Step 4: The remaining 4 positions must be filled with the 4 distinct letters L, A, D, R in some order. Since these letters are all different, they can be arranged in $4!=24$ ways.
Step 5: By the multiplication principle, total distinct arrangements $=15\times24=360$.
\[\boxed{360}\]