Is the questioner considered sinful for delaying the giving of Zakat al-Fitr until after Eid, and is it permissible to count what she spent in Ramadan as Zakat al-Fitr on behalf of her friend, considering the amount sent as sadaqah from her, and what is incumbent upon her to do now?

Zakat al-Fitr requires intention. Your payment of the amount on behalf of your friend with the intention of it being Zakat for her will be counted as such for her, and cannot be considered as Sadaqah (charity) from you. What you spent previously cannot be considered as Zakat for her.

Delaying the payment of Zakat al-Fitr beyond its specified time (from Fajr prayer until the Eid prayer) is not permissible, whether the Zakat is for yourself or on behalf of someone else, due to contradicting the command of the Prophet (peace be upon him) that it should be paid before people go out for prayer. However, if the delay is due to a legitimate excuse (such as the absence of an eligible recipient or someone to send it with), there is generally no sin in it. It is preferable not to distribute Zakat outside the country of the one paying it, except out of urgent necessity.