That is the unstressed accusative form of the third person, feminine, singular pronoun and it's just something that exists in Romanian.
First, let's replace "the woman" with "her" and we have:
"The orderlies are washing her" - "Infirmierele o spală pe ea."
We have the stressed ("o") and the unstressed ("ea") forms. There isn't an exact mapping to English, as there are two of them, but they clearly correspond to "her". The idea is that the unstressed form is mandatory, while the stressed one is optional, usually used for emphasis, or clarification. So we can say:
"Infirmierele o spală."
Also, as with the sentence at hand, the unstressed form is mandatory even when using a noun and it must match said noun's grammatical gender:
"The orderlies are washing the man." - "Infirmierele îl spală pe bărbat."
You can read more about the stressed/unstressed forms here.
Thank you, but I still don't understand why it is mandatory to include the pronoun here, even when using a noun, but not in earlier sentences like 'bărbatul mănâncă pasărea'? Isn't the bird being eaten by the man just as accusative as the woman being washed by the orderlies?
It's because of the definite article. Using it here, we would have:
Infirmierele spală femeia.
The "o" is gone. Similarly, removing it from your sentence, we would have:
Bărbatul o mănâncă pe pasăre. (but this sounds unnatural and I recommend against it.)
I'm sorry I didn't address this in my last comment, but I was thinking more about pronouns.
As to when/why you can or can not use the definite article is beyond me to try and formalize, I can only give you my opinion as a native speaker about what sounds natural and what doesn't.