Charlotte is right that "&c" means "et cetera".

Et Cetera, abbreviated to etc., etc, et cet., ect, **&c. or &c**is a Latin expression that is used in English to mean "and other similar things", or "and so forth".
http://en.m.wikipedia.org/wiki/Et_cetera

However, this is a woman writing to her mother, which is why she says "with best love".
http://www.lingq.com/en/learn-english-online/courses/51562/3...

The "&c" basically means "there are other salutations I could use here if I had more time". So, for example, she could say "I am your most devoted daughter".

I hope that explains it.

&c. is an abbreviation of etc., which is an abbreviation from the Latin words et cetera, meaning and others, or, and so forth; forth means onward or forward. Hence there can be no propriety in saying I am yours and others, or I am respectfully yours and onward.
https://english.stackexchange.com/questions/43403/meaning-of...
Other forms
"Yours aye" is a Scottish expression meaning "Yours always", still commonly used as a valediction to end written correspondence in the Royal Navy and British Army,[21] and occasionally used by sailors or people working in a maritime context. It is commonly used in the Royal Australian Navy as a sign-off in written communication such as emails.
"Yours, etc." is used historically for abbreviated endings. It can be found in older newspaper letters to the editor, and often in US legal correspondence. "&c." may be seen as an alternative abbreviation of et cetera, the ampersand functioning as a ligature form of "et". "I am, etc." and "I remain, etc." are also used.
"Yours hopefully" is occasionally used in letters of respect or complaint
https://en.wikipedia.org/wiki/Valediction#Other_forms

Many sign-offs in bygone days would have been something like Charlotte suggests "I am (or I remain) you obedient servant / himble admirer / and so on, according to the relative status of the 2 people.
Here it is just a lazy way of saying "... and all the rest of the usual waffle!"

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Note added at 46 mins (2022-05-21 13:40:00 GMT)
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Erm; admirers are more usually 'humble' — sorry!

The novel can be found here: https://www.gutenberg.org/files/946/946-h/946-h.htm

If you review the many instances of '&c.' in the novel you will discover a pattern of use which reflects the social pecking order of the letter-writers and their respective recipients; the degree to which the writer is trying to curry favour or support from the recipient; the author's expectation that his/her words will be regarded as authoritative; etc.

For example, where the letter-writer is a parent giving advice to one of their off-spring a flowery closing is considered unnecessary - the recipient knows his/her place without having it spelt out in full every time. And a lot of 'waffle' in the closing may even dilute the force of the message.

In contrast, when a child writes to one of their parents, or some other person of supposedly 'higher' social standing, or if they are asking for something, they do spell the closing out in full as a sign of deference to their 'superior'. Depending on the relationship, the flowery language used may be interpreted as either a sign of affection or respect - or, in some cases, as sarcasm or a way of putting someone 'in their place'.

The use of '&c.' should not be interpreted as if it were an abbreviation for any longer expression (Yours sincerely, etc.), because it is there to provide clues (some more subtle than others) as to the relationships between the different characters.