Metrizable group 
Let $ G $ be a metrizable group. If (i) $ K $ is a closed normal subgroup of $ G $ and (ii) both $ K $ and $ G/K $ are complete, then $ G $ is complete.

Here is how I am proceeding:
It can be assumed w.l.o.g. that the topology of $ G $ is induced by a right-invariant metric $ d $. Let $ (x_{n})_{n \in \mathbb{N}} $ be a right Cauchy sequence in $ G $. It suffices to show that some neighborhood $ V $ of $ e $ in $ G $ is complete. I don’t know how to proceed after this. Please help me.
 A: It seems that there is no need to prove anything, you can simply use the references. 
Lemma 1. [Eng, Th. 4.3.26] A topological space $X$ is metrizable by a complete metric iff $X$ is \v Cech-complete and metrizable.
Lemma 2. ([Bro] or [Cho]) A first countable $T_1$ topological group $G$ is Raikov complete iff $G$ is \v Cech-complete.
Tkachenko in [Tka] wrote that Vilenkin shows that the class of Weil-complete topological groups is closed under extensions: if $N$ is a closed normal of subgroup of a $T_1$ topological group $G$ and both groups $N$ and $G/N$ are Weil-complete, then $G$ is Weil-complete, and following Vilenkin’s reasoning, Graev [Gra] proves a similar result for Raikov-complete topological groups. 
Referenses 
[Bro] L.G. Brown, {\it Topologically complete groups}, Proc. Amer. Math. Soc. {\bf 35} (1972), 593--600.
[Cho] M.M. Choban, {\it On completions of topological groups}, Vestnik Moskov. Univ. Ser. Mat. Mekh. (1) (1970), 33--38  (in Russian).
[Eng] R. Engelking. {\it General Topology}. -- M.: Mir, 1986 (in
Russian).
[Gra] M.I. Graev, {\it Theory of topological groups I}, Uspekhy Mat. Nauk {\bf 5} (1950), 3—56 (in Russian).
[Tka] M.G. Tkachenko, {\it Topological groups for topologists: part I}, Bol. Soc. Mat. Mexicana (3), Vol. 5, 1999.
