Here's a sketch of a proof that the constant you want exists, and how to find it.

Let
$$
f(n) = \arctan(1) + \arctan(1/\sqrt{2}) + \arctan(1/\sqrt{3}) + \ldots + \arctan(1/\sqrt{n}).
$$
You want to show that $f(n) = \sqrt{n} + C + o(1)$ for some constant $C$. (If you're not familiar with the $o$-notation, think of $o(1)$ as representing some function which goes to $0$ as $n$ goes to infinity.)

Then take the power series expansion of $\arctan(1/\sqrt{k})$; this is

$$
(*) ~~~~~~~k^{-1/2} - \frac{1}{3} k^{-3/2} + \frac{1}{5} k^{-5/2} + \ldots
$$

So summing over $1$ to $n$, we should get
\begin{align*}
f(n) = & (1^{-1/2} + 2^{-1/2} + ... + n^{-1/2}) \\\
- \, \frac{1}{3} &(1^{-3/2} + 2^{-3/2} + ... + n^{-3/2}) \\\
+ \, \frac{1}{5}& (1^{-5/2} + 2^{-5/2} + ... + n^{-5/2}) - \ldots
\end{align*}
Now, $1^{-1/2} + 2^{-1/2} + \ldots + n^{-1/2}$ has the asymptotic form
$$
2 \sqrt{n} + \zeta(1/2) + O(n^{-1/2})
$$
where I cheated a bit and asked Maple, and $\zeta$ is the Riemann zeta function. And $1^{-j/2} + 2^{-j/2} + \ldots + n^{-j/2}$ has the asymptotic form
$$
\zeta(j/2) - O(n^{-j/2 + 1})
$$
where, if you're not familiar with the $O$-notation, $O(n^{-j/2+1})$ should be thought of as a function that goes to zero at least as fast as $n^{-j/2 + 1}$ as n goes to infinity. So, assuming that we can rearrange series however we like,
$$
f(n) = 2 \sqrt{n} + (\zeta(1/2) - \frac{1}{3} \zeta(3/2) + \frac{1}{5} \zeta(5/2) - \ldots) + o(1).
$$
Since $\zeta(s)$ is very close to $1$ when $s$ is a large real number, that alternating series should converge. Again cheating and using Maple, I claim it converges to about $−2.157782997$. This is the constant you call $\varphi$, and what you called $K$ is equal to $2$. (An easier way to see that your $K$ is $2$ is to note that $\arctan(1/\sqrt{n})$ is about $1/\sqrt{n}$, and approximate the sum by an integral.