I understand the 401k reference to be the U.S. pension contribution scheme. I understand that $139.04 is your withheld tax and 401k contribution for about two weeks' work.
Your work allowance of 1040 hours per year is 1000*10.25=$10250 plus 40*10.25=$10660 gross, as you said. By part-time I guess that means that over a year you can only work 1040 hours in total = 26 40-hour weeks, and when you are working your regular work hours are 40 hours a week. You haven't stated your normal weekly hours. You could, for example, work 20 hours a week for a year, because 20*52=1040 hours.
Your biweekly take-home pay of $820 implies that the gross amount is 820+139.04=$959.04, assuming your employer deducts tax withholding and 401k contribution before paying you. $959.04 is $479.52 a week or $11.99 an hour in a 40-hour week. Tax withholding and 401k contribution is 139.04/959.04=0.145 or about 14.5% of your gross pay. Therefore, if you know your gross biweekly pay, you can calculate the tax withholding and 401k contribution by multiplying the gross amount by 14.5%.
Multiplying the deductions by 0.005%, which is a very small amount, seems pointless, because you want the tax withholding and 401k contribution, which you already have! Dividing by 0.005% seems more appropriate, although it could only give you something related to your gross pay. Since 0.005% is equivalent 0.00005 you would end up with a figure 20000 times bigger, giving you an amount 20000*139.04 or around 278000 which is way in excess of your annual earnings! 14.5% of your annual earnings is 10660*0.145=$1545.70, or $59.45 biweekly, but your biweekly deductions are known to be $139.04, more than twice as much.
$820 is equivalent to 80 hours of work (over two weeks) at $10.25 an hour, so your work rates may have been quoted as what you take home after deductions have been made (net pay), rather than the gross amount you indicated.
There should be enough information in your payslips and your annual pay statement to work out how and what deductions are made before you receive your earnings, and exactly what your are paid gross by the hour. Perhaps your mother-in-law could also help to justify her calculations, particularly since I'm unfamiliar with the American tax system.
Can I offer another possible explanation and calculation that may fit your information better? Let's say your gross pay is $11.99 per hour, then for 1040 hours per year, you would receive $12469.60 gross pay. Divide this by 26 to get the biweekly wage: $479.60. Take out the deductions of 14.5% and that leaves 85.5% of $480=$410.06 and the deductions are $69.94. The deductions for $820.12 (2*$410.06) would therefore be 2*$69.94=$139.88 and the net hourly rate for a 40-hour week paying $410.06 would be $10.25. I'm not saying this is the true case, but if we change gross to net in your question, the figures seem to fit the facts better, don't you think?
