Its not quite clear why you needed to wait for BT to provide you the ISP information, you should have been able to get this from just looking up what ISP is assigned the IP range.
Check out
http://ripe.net/ for the future.
To identify the suspected individual you will need the authorities and ISP involved. To actually prove this, short of someone owning up I am sure a forensic investigation will be involved.
You have not really suggested what is supposed to of happened, aside from someone accessing a networked machine via logmein.