Nielsen & Chuang Theorem 2.6 Proof

I got a problem in understanding the proof of the Theorem 2.6 (Unitary freedom in the ensenble for density matrices), 2.168 and 2.169 in the Nielsen and Chuang book

Equation 2.168

Suppose $|{tildepsi_i}rangle = sum_j{u_{ij}|{tildevarphi_j}rangle}$ for some unitary $u_{ij}$. Then
$sum_i{|{tildepsi_i}ranglelangletildepsi_i|} = sum_{ijk}{u_{ij}u_{ik}^*|tildevarphi_jranglelangletildevarphi_j|}$ (2.168)

I don’t get this step. If I take $langletildepsi_i|=(|tildepsi_irangle)^dagger=sum_j(u_{ij}|tildevarphi_jrangle)^dagger=sum_j{langletildevarphi_j|u_{ij}^dagger}$ and substitute this in the outer product I receive $sum_i{|{tildepsi_i}ranglelangletildepsi_i|} = sum_{ijk}{u_{ij}|tildevarphi_jranglelangletildevarphi_j|u_{ik}^dagger}$
Can someone explain this to me please?

Equation 2.169 -> 2.170
$$sum_{jk}{(sum_i{u_{ki}^dagger u_{ij})}|tildevarphi_jranglelangletildevarphi_k|} = sum_{jk}{delta_{kj}|tildevarphi_jranglelangletildevarphi_k|}$$
I can’t understand why $(sum_i{u_{ki}^dagger u_{ij}}) = delta_{kj}$.
I understand that $u_{ki}^dagger u_{ij} = I$ for $k=j$, but why is it zero otherwise?

It would be so kind if someone could enlighten me.