So there are two bugs:
The values are not properly scaled. Each number should be around one tenth of their current values. In fact, if the number of observation increases by 10^n n = 1, 2, ... then the cdf also increases by 10^n
For example:
> kernelDensity(rand(1000))
> 953.53
They don't sum up to 10 (or one if it were not for the scaling error). The error becomes more evident as the sample size increases: there are approx. 5% of the observations not being included.
I believe that I implemented the formula 1:1, hence I really don't understand where the error is.