Trouble finding ##L^2## in function of ##x## and ##p##

[pixyl]

Homework Statement

Prove that $\vec L^2 = \vec x^2 \vec p^2 - (\vec x \cdot \vec p)^2 + i\hbar \vec x \cdot \vec p$

Relevant Equations

$\varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}$,
$L_i = \varepsilon_{ijk}x^jp^k$,
$[x^i, p^j] = i\hbar\delta_{ij}$
$[x^i, x^j] = [p^i, p^j] = 0$

What I've done is
$$\vec{L}^2 = \varepsilon_{ijk}x^jp^k\varepsilon_{imn}x^mp^n = (\delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km})x^jp^kx^mp^n = x^jp^kx^jp^k - x^jp^kx^kp^j = $$
$$ = x^jx^jp^kp^k - i\hbar x^jp^j - x^jp^kx^kp^j = $$
$$ = x^jx^jp^kp^k - i\hbar x^jp^j - (x^jx^kp^kp^j - i\hbar x^jp^j) = $$
$$ = x^jx^jp^kp^k - i\hbar x^jp^j - (x^jx^kp^jp^k - i\hbar x^jp^j) = $$
$$ = x^jx^jp^kp^k - i\hbar x^jp^j - x^jp^jx^kp^k = $$
$$ = \vec x^2 \vec p^2 - (\vec x \cdot \vec p)^2 - i\hbar \vec x \cdot \vec p$$

I've tried again and again and I don't understand what I'm doing wrong: if you can spot the error it would be greatly appreciated.

 

[vela]

$p_k x_k=x_k p_k - i\hbar \delta_{kk} = x_k p_k - 3i\hbar$

 

[pixyl]

$p_k x_k=x_k p_k - i\hbar \delta_{kk} = x_k p_k - 3i\hbar$

Oh my god yes... I just thought $\delta_{kk} = 1$ so I ignored it, forgetting about the sum. Thank you, you've ended my two days misery!