Take out the hard drive and put it in your new system.
... then the PS3 asks to format the HDD.
The PS3 encrypts the HDD and it only works in the console it is used unless formatted and encrypted again for the new system.
It's not possible to just put it in the new system.
....buy PlayStation + membership -> make backup files and save it to the web -> put you brand new 160 back -> load your backup files from the web and

.
or simply a USB memory stick? No need for PSN+ to backup saves. PSN+ is only needed if the saves were copy protected.
But without access to the save files it either way not possible.
but how the hell am i going to get my level 31 sneaky khajit on to the new system, this is the new dilemma.
I don't see a way how this might be possible.
Since the PS3 is broken and doesn't start you can't access the save files in the first place and therefore no way to backup them
