installing snow leopard w/ time machine and no desk.

No.
You need some kind of install DVD, thus you need the grey Restore DVDs (contact Apple for those, they send them to you for a small fee depending on your country) if you have a 2010 or 2011 iMac, or buy the 29 USD Mac OS X 10.6 Snow Leopard Upgrade DVD from the Apple Online Store if you have a 2009, 2008, 2007 or 2006 iMac.

 

While it won't help you this time because you now know you need an Install Disc what you may want to consider for the future is getting a second external HD and using it as a bootable clone. A bootable clone is just that, in the event of a HD failure you can boot and run from the clone, in addition if you have another HD failure you can boot from it, then format the replacement internal HD and install from the clone! While Time Machine is very useful it's not without it's shortcomings, many people that are serious about keeping backup do multiple backups because backup can fail too! So redundancy in the case of backup strategies is actually a very good thing. If you are interested in creating a bootable clone both SuperDuper and Carbon Copy Cloner applications will do this for you and are pretty easy to use.