Now we want to find out whether we should pay zakaah on this gold for this year only or for the other years for which we are not sure about her zakaah. Will this discharge her obligations if she fell short in working out the amount of zakaah because of some lack of knowledge on her part? Thank you very much.
There is nothing wrong with paying the zakaah on jewellery on the basis of estimation, if the person thinks it most likely that he has paid the amount required. For more information please see the answer to question no. 145091.
You mentioned in your question that she used to pay zakaah, but you are not sure about the manner in which it was paid. If that is the case, then the basic principle is that she was paying zakaah in the manner which would discharge her obligations, and your uncertainty does not affect this principle.
The basic principle is that if the Muslim has performed an act of worship, then it is valid and acceptable unless there is any proof to the contrary.
Based on that, the heirs do not have to pay zakaah on the jewellery for the past years. But if you want to do an act of kindness towards her and give some money in charity on her behalf, then the reward of that will reach her, in sha Allah.
And Allah knows best.