How to help an old lady?  

Maybe tell her honestly in a sweet manner or you can buy newer clothes that looks like what she's wearing, maybe her clothes has sentimental value that's why she doesnt take them off :)

Take her some bath powders. Not perfume, but actual bath powders. You should be able to buy them at Wal-Greens or any other drug store.

Tell her someone gave them to you and you are giving them to her. Tell her they make you sneeze or something. Or buy her some scented lotion and say the same thing.

As for the clothes, that one is tougher and I have no suggestions.

Why not just accept her as she is. You have obviously looked beyond the dirt and smell to see someone worth knowing. Keep it that way. Just be her friend for now. There may be a time when you can help in the other areas that bother you but don't force it. You made the offer to help once and she gave you her answer, accept it. Some people are too proud to accept help or are ashamed of their circumstances.

some people never change. they have their own ideas and sentiments. The old lady youe mentioned has an experience of years which you wont get. there must be some valid reasons which she may not tell to any one.

Oh, that's very sweet and kind of you. Maybe you can make her change those clothes away by asking her if she want to go somewhere that she'd really like to go then tell her that before you can go there she must take a bath first or atleast change her clothes so she can look presentable. Or maybe offer clothes that you think she'll like that'll make her wear it. Ask her favorite color things like that. Or you can tell her that you are going to a friend's house and you want to bring her with you but before you both go there you must change your clothes because its a costume party or she needs to wears clothes like this or that, just to make her take off those dirty clothes. Or maybe you can just change her clothes while she's sleeping, if you think that is possible.

I think you should tell her that people were talking that she looks dirty & people were looking her like she was a mentally upset due to her dirty clothes. try to convince her.

Maybe buy her some new clothes and say it is a gift or something, say you think of her as a grandmother and that it is a late christmas gifts?

its like a child with thier favorite blanket
some old people kind of revert back to a childlike state and so thier insecurities get locked up into cetain things that sooth them
i am actually the same way as a child i had a favorite blanky that i dragged around with me everyhere it stank ,was dirty but it felt so god and made me comfortable ,now i have a old nightgown i have worn foreer and same it is worn out ,as long as no one but my family sees it i wear it but if friends come over i wear a new one.
maybe she just cant be without her comfort zone around her

when u think of helping her u can definitely tell her that how she is to others .there is nothing wrong.as she is good at character u can say her to b good at dressing too.actually she is alone so may b she dont cares of herself.u can tell her to b so.

if u r so close to her, then its better u dont hide anything 4m her. be frank to her n say wht ever u feel in a pleasing manner. buy a reasonable deodrant for her n if u really like her then ask her to use them regularly as they keep her health in gud condition. in these ways u cud pretend her to be gud n i hope even she cud understand u n make her self neat n tidy all time.