Another Modern English cognate even closer to shyne than "sheen" is "shine" (and obviously the German "schein"). The words for "beautiful", "fair", "bright", "shining", "well-reputed", "righteous" have a long history of being related:

https://en.wiktionary.org/wiki/schinen#Middle_English (to shine, to appear)

https://en.wiktionary.org/wiki/skyr#Middle_English (clear-coloured, pale, light, luminous, radiant)

https://en.wiktionary.org/wiki/sciene#Old_English (beautiful, fair, brilliant, shining)

https://en.wiktionary.org/wiki/Reconstruction:Proto-Germanic... *skīnaną (to shine, to appear)

https://en.wiktionary.org/wiki/Reconstruction:Proto-Germanic... *skīriz (pure, clear, sheer)

https://en.wiktionary.org/wiki/Reconstruction:Proto-Germanic... *skauniz (beautiful, shining)

and ultimately the PIE

https://en.wiktionary.org/wiki/Reconstruction:Proto-Indo-Eur... *(s)ḱeh₁y- (to shine)

There are cognates absolutely everywhere in modern Germanic languages:

https://en.wiktionary.org/wiki/sk%C3%ADr#Icelandic skír (bright, clear, pure)

https://en.wiktionary.org/wiki/skir#Swedish (sheer, delicate, shining)

And even in Slavic languages:

https://en.wiktionary.org/wiki/Reconstruction:Proto-Slavic/s... *sijati (to shine, to illuminate)

Skauniz was even borrowed to Proto-Finnic and highly conserved in modern Finnish, Estonian, Ingrian, etc. which all have kaunis meaning "beautiful"!

https://en.wiktionary.org/wiki/Reconstruction:Proto-Finnic/k... *kaunis

One way to say it that is understandable in modern English and Swedish: She shines with beauty/ Hon skiner av skönhet

This makes sense in the Firefly universe, too. Shiny!