They should've done a chapter in her life, like "My Week with Marilyn." In that film, you get a glimpse of her life, and who she was privately and publicly and on the set. In fact, I think the story told in "My Week with Marilyn" is ripe for musicalization. The movie is told through the eyes of Colin Clark, a gofer on set who becomes smitten with Marilyn and eventually protective of her. [This is one thing many people who met her confirm, and Marilyn herself often took advantage of this.]
Anyway, a lot of musicals require a narrator of sorts, and Colin Clark fits the bill. Through him we get an inside look not only into the making of "The Prince and the Showgirl," but also into Marilyn herself: her personal struggles, insecurities, stage fright, neediness. The story also has a cast of great supporting characters, like Laurence Olivier (director and co-star, who becomes exasperated with Marilyn's tardiness/absences and overall lack of professionalism), Dame Sybil Thorndike (legendary British actress and Marilyn's champion on set; comic relief), Paula Strasberg (Marilyn's meddling acting coach and sycophant, who causes further conflict between Olivier and Marilyn), and Arthur Miller (Marilyn's distant and regretful 3rd husband, who is not and does not want to be the rock Marilyn so desperately wanted and needed).
And of course at the center of it all we have Marilyn -- at times flirtatious and seemingly happy-go-lucky, and at other times desolate and forlorn, but all the time luminous. The part of Marilyn would be a great part for any actress 'cause she's not one-note. Her emotions run the gamut.
Salve, Regina, Mater misericordiae
Vita, dulcedo, et spes nostra
Salve, Salve Regina
Ad te clamamus exsules filii Eva
Ad te suspiramus, gementes et flentes
O clemens O pia