The series can be dominated by $$ b_{k,n} \le 2^{-\frac{k}{2}}$$
For $n < k$ this is obivously true, as we have $b_{k,n} = 0$. For $n\ge k$ we have
$$ n \ge k$$
$$ \frac1n \le \frac 1k$$
$$ \frac1k + \frac1n \le \frac2k$$
$$ \frac{kn}{k+n} = \frac{1}{\frac1k + \frac1n} \ge \frac{k}{2}$$
$$ b_{k,n} = 2^{-\frac{kn}{k+n}} \le 2^{-\frac{k}{2}}$$
Series $ \sum_{k=0}^\infty 2^{-\frac{k}{2}} = \sum_{k=0}^\infty \frac{1}{(\sqrt{2})^k}$ is convergent, so from the dominated convergence theorem, you can justify the swapping of the sum and the limit.
More fundamental proof:
Let us decompose $$ \sum_{k=0}^n 2^{-\frac{kn}{k+n}} = \sum_{k=0}^{\lfloor\sqrt{n}\rfloor} 2^{-\frac{kn}{k+n}} + \sum_{k={\lfloor\sqrt{n}\rfloor+1}}^n 2^{-\frac{kn}{k+n}}$$
For $k \in [\lfloor\sqrt{n}\rfloor + 1,n] $ we have $$ \frac{kn}{k+n} > \frac{\sqrt{n} n}{n+ n}= \frac{\sqrt{n}}{2}$$ $$ 2^{-\frac{kn}{k+n}} < 2^{-\frac{\sqrt{n}}{2}}$$
We have then $$ \sum_{k={\lfloor\sqrt{n}\rfloor+1}}^n 2^{-\frac{kn}{k+n}} < (n-\lfloor\sqrt{n}\rfloor) 2^{-\frac{\sqrt{n}}{2}} < n2^{-\frac{\sqrt{n}}{2}} \rightarrow 0$$
so
$$ \lim_{n\rightarrow\infty} \sum_{k={\lfloor\sqrt{n}\rfloor+1}}^n 2^{-\frac{kn}{k+n}} = 0$$
For $k \in [0,\lfloor\sqrt{n}\rfloor]$ we have $$ \frac{kn}{\sqrt{n}+n}\le \frac{kn}{k+n} \le k $$
$$ 2^{-k} \le 2^{-\frac{kn}{k+n}} \le 2^{-\frac{kn}{\sqrt{n}+n}} $$
$$ \sum_{k=0}^{\lfloor\sqrt{n}\rfloor} 2^{-k} \le \sum_{k=0}^{\lfloor\sqrt{n}\rfloor} 2^{-\frac{kn}{k+n}} \le \sum_{k=0}^{\lfloor\sqrt{n}\rfloor} 2^{-\frac{kn}{\sqrt{n}+n}} $$
$$ \frac{1-2^{-\lfloor\sqrt{n}\rfloor-1}}{1-2^{-1}} \le \sum_{k=0}^{\lfloor\sqrt{n}\rfloor} 2^{-\frac{kn}{k+n}} \le \frac{1-2^{-\frac{(\lfloor\sqrt{n}\rfloor+1)n}{\sqrt{n}+n}}}{1-2^{-\frac{n}{\sqrt{n}+n}}} $$
We have $$ \lim_{n\rightarrow\infty} \frac{1-2^{-\lfloor\sqrt{n}\rfloor-1}}{1-2^{-1}} = \lim_{n\rightarrow\infty} \frac{1-2^{-\frac{(\lfloor\sqrt{n}\rfloor+1)n}{\sqrt{n}+n}}}{1-2^{-\frac{n}{\sqrt{n}+n}}} = 2 $$
which means that also $$ \lim_{n\rightarrow\infty} \sum_{k=0}^{\lfloor\sqrt{n}\rfloor} 2^{-\frac{kn}{k+n}} = 2 $$
Altogether we have $$ \lim_{n\rightarrow\infty} \sum_{k=0}^n 2^{-\frac{kn}{k+n}} = \lim_{n\rightarrow\infty} \Big(\sum_{k=0}^{\lfloor\sqrt{n}\rfloor} 2^{-\frac{kn}{k+n}} + \sum_{k={\lfloor\sqrt{n}\rfloor+1}}^n 2^{-\frac{kn}{k+n}}\Big) = 2 + 0 =2$$
