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)) + ... + 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) - 1/3 k^(-3/2) + 1/5 k^(-5/2) + ...

So summing over 1 to n, we should get

f(n) = (1^(-1/2) + 2^(-1/2) + ... + n^(-1/2))
- 1/3 (1^(-3/2) + 2^(-3/2) + ... + n^(-3/2))
+ 1/5 (1^(-5/2) + 2^(-5/2) + ... + n^(-5/2))
- ...

Now, 1^(-1/2) + 2^(-1/2) + ... + 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. Zeta is the Riemann zeta function. And 1^(-j/2) + 2^(-j/2) + ... + 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) - 1/3 Zeta(3/2) + 1/5 Zeta(5/2) - ...) + 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 φ, 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.