Fabrizio-

For weighted least squares, the idea is to minimize the sum of squares of the random factors of the estimated residuals, Q, with respect to beta.  (dQ/db = 0 should do that, using b for "beta" here, though b is technically the estimate of beta, but we are actually estimating beta.  Also, this is a partial derivative, but symbols are limited on my phone.)

Below are links to examples of that process:

Pages 2 - 4 here for one regressor and zero intercept, and pages 13 - 15 with an intercept term:

 https://www.researchgate.net/publication/263036348_Properties_of_Weighted_Least_Squares_Regression_for_Cutoff_Sampling_in_Establishment_Surveys

For two regressors, zero intercept, see pages 597 - 598 in the following:

 https://www.researchgate.net/publication/261534907_WEIGHTED_MULTIPLE_REGRESSION_ESTIMATION_FOR_SURVEY_MODEL_SAMPLING

You just set the partial derivative of the sum of squares of the random factors of the estimated residuals with respect to beta, equal to zero, and solve to estimate beta. 

Cheers - Jim

PS - I can't open your file on this phone.