She is referred to as the 'Gyalsem', I assume that means 'princess' as her brothers are referred to as Gyalsey (prince), her mother is the (Druk) Gyaltsuen ((dragon) queen) and her father is the (Druk) Gyalpo ((dragon) king). However, on wikipedia the words 'ashi' and 'dasho' are used in front of their names...
The young princess shares her name with one of her aunts: Sonam Dechen Wangchuck (half-sister of the king by the fourth king's first wife). Her grandmother is Tshering Yangdon, so her second name (Yangden) seems a variant on her grandmother's name.
Her brothers also shared their name with close family members: Jigme of course with their father, grandfather and some of their uncles and cousins), Jigme Namgyel also shares his second name with his father. Jigme Ugyen shares his second name with one of his uncles: Ugyen Jigme Wangchuck (half-brother of the king by the fourth king's first wife) and two cousins: Ugyen Junay Wangchuck (second son of his half-aunt Kesang Choden Wangchuck - by second wife) Ugyen Dorji Wangchuck (son of Dechen Yangzom Wangchuck, the king's only full sister); and his full name with another cousin: Jigme Ugyen Wangchuck (son of his eldest half-aunt Chimi Yangzom Wangchuck - by second wife).
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The little Princess' name has been revealed: 
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