Here's a way to get a time of 

>!12 minutes

Arranges conversations by 

>! spliting the party into two groups of five, one with Al and one with Jane, and have each group split into 2+3 in each of the ten possible ways.

You can get another two minutes by

>! Having each group of 5 in a conversation.

and 

>! Doing the same but splitting a person from each group (not Jane) into a conversation pair, leaving each half with 4.

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Here's a way to get a slighly better time of 

>! 13 minutes

There's nothing that systematic here. I'll just post the list of conversation clusters. Each line list the people who know the secret beforehand, followed by the conversations, with secret-sharing conversation before the semicolon.

>! <code>
>! A:   AB; CDEFGHIJ
>! AB:  ABC; DEFGHIJ
>! ABC: ABCD; EFGHIJ
>! A-D: AD BC; EGI FHJ
>! A-D: AC BD; EGJ FHI
>! A-D: ABCDE; FGHIJ
>! A-E: AE BCD; FG HIJ
>! A-E: BE ACD; FH GIJ
>! A-E: CE ABD; FI GHJ
>! A-E: ABCDEF; GHIJ
>! A-F: ABE CDF; GH IJ
>! A-F: ACE BDF; GI HJ
>! A-F; BCE ADF; GJ HI
>! </code>