How to find the public IP within IOS

You can't do that, but you can configure the router so it prompts for name/password at login, and based on the user's privilege, they will be in enable mode or not. So instead of a two-step login, they login once and are put in the proper mode.

username farouk password foo 
username jong-mee password bar privilege 15

aaa new-model
aaa authentication login default local
aaa authorization enable default local

When jong-mee logs in, she gets prompted for her password, and immediately goes into enable mode. But when Farouk logs in, he is only in user mode.

Make the enable secret long and complicated so it can't be guessed. You won't have to use it anymore.

The solution you suggest might work as long as you do not need to do NAT on the Cisco. You can simply configure eg 192.168.1.10 on the interface as follows:

FastEthernet X ip address 192.168.1.10 255.255.255.0 secondary

Then add a default route towards your VDSL router. Do not forget to also add a static route for you internal network to the VDSL router via the Cisco. If that is not possible you need to do NAT...

Another solution would be to define a second VLAN, eg. VLAN 2 on the Cisco and place one of the switchport Interfaces into that VLAN and the route between these two interfaces as above. This would also enable you do do NAT.

But the relay easiest way would be to just reconfigure your VDSL router's LAN IP to be 192.168.10.236 and disable the DHCP server on it, then connect it to your switch, magic done.