Albert chooses a positive integer $k$, then two real numbers $a,b$ are randomly chosen in the interval $[0,1]$ with uniform distribution.
The square root of the sum $(k\cdot a+1{)}^2+(k\cdot b+1{)}^2$ is then computed and rounded to the nearest integer. If the result is equal to $k$, he scores $k$ points; otherwise he scores nothing.

For example, if $k=6$, $a=0.2$ and $b=0.85$, then $(k\cdot a+1{)}^2+(k\cdot b+1{)}^2=42.05$.
The square root of $42.05$ is $6.484\cdots$ and when rounded to the nearest integer, it becomes $6$.
This is equal to $k$, so he scores $6$ points.

It can be shown that if he plays $10$ turns with $k=1,k=2,\dots ,k=10$, the expected value of his total score, rounded to five decimal places, is $10.20914$.

If he plays ${10}^5$ turns with $k=1,k=2,k=3,\dots ,k={10}^5$, what is the expected value of his total score, rounded to five decimal places?