Assume that the range of dummy indices is from 1 to N
$$\delta_{ij} \delta_{jn} = \delta_{i1} \delta_{1n} + \delta_{i2} \delta_{2n} + \delta_{i3} \delta_{3n} +\cdots + \delta_{ii} \delta_{in} + \cdots \delta_{in} \delta_{nn} + \cdots + \delta_{iN} \delta_{Nn} $$
Assuming
$$i = n$$
we get
$$\delta_{ij} \delta_{jn} = \delta_{ii} \delta_{in} + \delta_{in} \delta_{nn}$$
Substituting i = n, we have
$$\delta_{ij} \delta_{jn} = \delta_{ii} \delta_{ii} + \delta_{ii} \delta_{ii} = 2$$
I know this is wrong and $$\delta_{ij} \delta_{jn} = \delta_{in}$$ but I don't know where I am making a mistake.