Is it permissible to repeat the circumambulation (Tawaf) and the Sa'i for 'Umrah if the mother was preoccupied with her children, distracting her from devotional remembrance (dhikr) and humility (khushu') during her performance, whether this occurred before or after the 'Umrah ritual disengagement (tahallul), and whether by forming a new intention from the hotel?

Your circumambulation (Tawaf) and striving (Sa'i) are valid, and you are not required to repeat them merely because you were distracted from devotion (khushu') and remembrance (adhkar). It is not prescribed to repeat them as a recommendation for that reason. Allah Almighty has prescribed one circumambulation and one striving for Umrah, and you have performed them. Repeating them without a valid legal reason might fall into the category of innovation (bid'ah). You can, however, increase voluntary circumambulations in Mecca with devotion and abundant remembrance.