You may not be able to get this to work... put simply, there are three main types of formats for e-mail messages:
Plain Text
HTML
RTF (Rich Text Format)
Plain Text is the most "compatible" format, in that every mail server on the Internet can pass it without problems. HTML is the second-most compatible format... sometimes, HTML messages must be converted into Plain Text in order for them to be passed. And finally, RTF is the least compatible format... RTF messages are often converted into HTML or even Plain Text when they pass through the Internet.*
When messages are converted from one format to another, anything within the message that's not text (such as the graphic in your signature) may be LOST, or it may be converted into an attachment. (Plain Text messages can contain attachments, but no embedded graphics or other non-text objects.)
It sounds like when the message is going from you to your friend, one or more mail servers through which the message is passing is converting the message into a different format, thereby resulting in the odd attachment behavior you're seeing. It's impossible for any of us to determine the cause of this... you'll need to work with your friend to see exactly what's going on. Try sending some messages in different formats (Plain Text, HTML, RTF) and see if you can isolate the behavior (e.g., if I do THIS, THAT happens). For example, see what happens to the message format for each type of message you send to her; you may find that you are sending RTF and it's not working, but HTML *might* work.
Good luck.
* Sending e-mail is not like sending a letter through the postal service. With e-mail, each server that passes the message "opens" it, "reads" it, and then "closes it up" and "forwards" it on. This can result in messages being altered while in transit.