Kindle Lending Library without a Kindle?

I currently have an iPad, iPhone and read using the Kindle app for the desktop while at work. I also have an Amazon Prime account.

If I purchase a cheap and broken old kindle and associate it with my Amazon account would I be able to borrow books from the Kindle Lending Library and read them on other devices?

No. Kindle Owner's Lending Library items can only be read on a Kindle device. I have a Kindle Fire as well as a Nexus 7. I can only read the KOLL books on the Fire.

It does sound like you must have a physical Kindle device to use the Amazon Kindle lending library. You also have to pay $80/yr for Kindle Prime subscription (that is what includes the lending library feature as well as 2nd day mail on all shipments and some movies/TV shows you can watch for free).

However - if you are in the US, check to see if your local library system supports Overdrive ebook lending service. Overdrive works not only on Kindle and Nooks but also with their software on tablets, PC and Mac. And you can borrow audiobooks too (modern Nook and Kindle eInk ereaders have no audio support but the other Overdrive reader options do).

Best - I believe this is a free service. You do need a valid library card and PIN from a local library that supports Overdrive and I believe this is available only in the US.
Scroll down a bit and on the right under the banner image search for your zip code to find local libraries that support Overdrive.
https://search.overdrive.com/

And here is a rundown of the the devices they support and their own reader apps.
http://www.overdrive.com/drc/

If you use your Kindle as the reading device, I believe the Overdrive books will be delivered and returned via your Manage Your Kindle page on Amazon (where you can choose which of your Kindle devices you want to send it to). You are contending for limited electronic loan copies so I think you queue/hold your choices (like Netflix) and when one is available for loan their notify you - you have 3 days to download it. The amount of time you can keep the book depends on the book.

I'm going to go get a new library card and a pin so I can borrow books to my Kindle.