There is also an issue with Eowyn that hasn't been explored as much as it should. Her uncle made her regent of the kingdom in his absence and that of her brother. Ruling Rohan in the place of Theoden was, I'm sure, a serious responsibility. (I'm presuming here that she exercised
real power, and wasn't a nominal regent, with an 'advisor' or 'advisors' exercising the real power.)
Due to this, I've had an issue over the last few years with her deserting her post as regent to fight in Gondor. If, for example, an ordinary Rider ordered to stay in Rohan behaved the same way she did, he would quite rightly be seen as having deserted his post and disobeyed orders, and would suffer serious disciplinary action. Eowyn, however, did not.
While she is a member of the royal family, this would add to the obligation to behave properly, to set a standard of behaviour for the people. My own view is that she was very lucky in fighting the Witch-king and being very seriously wounded. This was, presumably, seen by many as a sufficient punishment for her desertion.
If that hadn't happened, and had she survived the Battle of Pelennor Fields slightly wounded or unscathed, I've a strong suspicion that either her uncle or brother would have had the unenviable task of sitting in judgement on her actions.
What do people think?