Is it permissible for my mother—who is unable to prostrate, so she bows while seated on a chair—to sit for a few seconds after rising from the second prostration and before standing for the next rak'ah, without saying "Allahu Akbar" between the prostration and the sitting, and then saying "Allahu Akbar" after getting up from the chair to stand?

The fundamental principle of prayer is to perform it in the manner of the Prophet, peace and blessings be upon him. If one is unable, they should do what they can. If the mother is unable to prostrate due to knee pain, there is no harm in her omitting it and suffice with gesturing while sitting. As for your statement that she sits for a few seconds on the chair when rising from the second prostration, and then stands for the next rak'ah, this resembles the jalsat al-istirahah (rest session). It is recommended that the takbir begins when one starts to raise their head from prostration and ends when the rest session is complete. Then, one rises to stand without takbir. The mother's past prayers are valid, but she should be made aware of the recommended manner.