My 6-year old MBP17 just lost its logic board. It is now officially a paper weight. My Q: Can I boot-up an old Mac Mini from the Time Machine backup created from my laptop (until I purchase a new MBP) ?
What OS were you using on the MBP? If it was Lion you can boot from a TM backup disk but all it will give you is the TM version of the Lion Recovery HD. Which won't be much help is using any programs or files in the TM backup.
You might want to consider taking the 6-year old MBP17 hard drive and use it as an external USB Drive . Then you could migrate your data to the new box and eventually use it as a bootable clone for backup.