When you raise/lower indices using the metric, you are not simply "multiplying by 1," i.e. inserting $(1/4){\delta_m}^m$. For instance, if you start with a vector field $X^\mu$, then lowering the index with the metric turns it into a one-form $X_\mu = g_{\mu\nu}X^\nu$. These objects are not the same, so you cannot get from one to the other just by multiplying by 1.
In your question, you want to both raise and lower a given index to go from
$${\Gamma^k}_{ij} {A_k}^j$$
to
$$\Gamma_{kij}A^{kj}.$$
To do this, we apply the rule for raising and lowering $k$ one step at a time:
$$\Gamma_{kij} = g_{kl}{\Gamma^l}_{ij},$$
and
$$A^{kj} = g^{km}{A_m}^j.$$
Combining these gives
$$\Gamma_{kij}A^{kj} = g_{kl}g^{km}{\Gamma^l}_{ij}{A_m}^j = {\delta_l}^m {\Gamma^l}_{ij}{A_m}^j = {\Gamma^l}_{ij}{A_l}^j.$$
The key step is to not use the same letter to label the dummy indices in the two contractions with $g$.