Remote VPN ! site-to-site

Question

Remoe VPN client cannot get across L2L (site-to-site) tunnel after making connection.
Topology:
[remote]->[ASA1]-><L2L}->[ASA2]->LAN2
The problem is at the remote client, which is using Cisco VPN client.
Remote client connection is made fine to [ASA1].
Problem is that remote client does not know route to network LAN2 and dumps traffic off to its default gateway rather than directing it to [ASA1] for forwarding to [ASA2]. ([ASA1] and [ASA2], of course, know about each other.)
Cisco VPN client has capability of being "told" subsequent routes (Status->statistics->Route details).
As I see it, the client must get this info from the ASA to which it makes its remote VPN connection.
The advice I am hoping for is the CLI or ASDM syntax I need to apply to get the ASA to provide this route information.
TIA

Answer 1

Adam, thank you for the comprehensive reply ... unfortunately it's not working.
1. The statements you list above were already there to facilitate the L2L.
2. I turned-off split tunneling (or think I did) and ran a test ... no joy.
This took me back to my original premise that the remote client doesn't know how to send the traffic (bound for L2L) down the remote tunnel and dumps it of to its default gateway (to the WWW).
If you're willing to look at it, I have attached screen shots of the client ipconfig and the Cisco VPN client - showing its routes.
The ipconfig seems to say that the remote connection has its default gateway, and the tunnel has none.
The VPN client screen shows it knows a route (192.168.5.0/24) to the ASA, but nothing beyond. The ASA does, in fact, know about the network (10.64.0.0/16) at the other end of the L2L.
As I see it, if I can find a way to get the ASA to advertise this route to the VPN client, the problem might be solved. The client will then know to forward the traffic to the ASA instead of dumping it to the default gateway.
TIA