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.

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.

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.

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