1. We start with the equation $a^2 + \sqrt{2017 - b^2} = k^2$ where $k$ is an integer. For this to be true, $\sqrt{2017 - b^2}$ must also be an integer. Let $\sqrt{2017 - b^2} = m$ where $m$ is an integer. Therefore, we have:
$$2017 - b^2 = m^2$$
This can be rewritten as:
$$2017 = b^2 + m^2$$
This implies that $b^2 + m^2 = 2017$.

2. We need to find pairs $(b, m)$ such that both $b$ and $m$ are integers. We test values of $b$ to find corresponding $m$ values that satisfy the equation.

3. Testing $b = 44$:
$$2017 - 44^2 = 2017 - 1936 = 81$$
$$\sqrt{81} = 9$$
So, one pair is $(b, m) = (44, 9)$.

4. Testing $b = 9$:
$$2017 - 9^2 = 2017 - 81 = 1936$$
$$\sqrt{1936} = 44$$
So, another pair is $(b, m) = (9, 44)$.

5. Now, we have two pairs $(b, m)$: $(44, 9)$ and $(9, 44)$. We need to find corresponding $a$ values such that $a^2 + m = k^2$.

6. For $(b, m) = (44, 9)$:
$$a^2 + 9 = k^2$$
This can be rewritten as:
$$k^2 - a^2 = 9$$
$$(k - a)(k + a) = 9$$
The factor pairs of 9 are $(1, 9)$ and $(3, 3)$. Solving these:
- For $(1, 9)$:
$$k - a = 1 \quad \text{and} \quad k + a = 9$$
Adding these equations:
$$2k = 10 \implies k = 5$$
Subtracting these equations:
$$2a = 8 \implies a = 4$$
So, one solution is $(a, b) = (4, 44)$.

7. For $(b, m) = (9, 44)$:
$$a^2 + 44 = k^2$$
This can be rewritten as:
$$k^2 - a^2 = 44$$
$$(k - a)(k + a) = 44$$
The factor pairs of 44 are $(1, 44)$, $(2, 22)$, and $(4, 11)$. Solving these:
- For $(2, 22)$:
$$k - a = 2 \quad \text{and} \quad k + a = 22$$
Adding these equations:
$$2k = 24 \implies k = 12$$
Subtracting these equations:
$$2a = 20 \implies a = 10$$
So, another solution is $(a, b) = (10, 9)$.

8. The solutions $(a, b)$ are $(4, 44)$ and $(10, 9)$. The sums $a + b$ are:
$$4 + 44 = 48$$
$$10 + 9 = 19$$
Therefore, the total sum is:
$$48 + 19 = 67$$

The final answer is $\boxed{67}$