Most affiliate merchants/networks have their own tracking system and won't be willing to integrate with yours (Google Analytics or other). So you have to do your own integration, this analysis is best done in something like Excel because of ease and accessibility.

So grab this data from your own web analytics software:
Clicks on affiliate links (use eventTracking in GA)

And grab this data from Affiliate Mechants:
Clicks recorded on their end
Number of sales

And mash it all up in Excel (you can add earlier funnel steps from your own analytics like CTR of affiliate links, etc).

Two problems:

1. Clicks recorded in GA will always be different to Clicks recorded by Affiliate Merchants due to different way of tracking (GA tracks unique clicks with event tracking, Merchants may track all clicks). This is an imperfection you have to live with and it's bearable as long as you make sure you've minimised all reasons why clicks may not be recorded on your end and on their end. 

Then pick one of them as the connecting factor between the two sets of data and stick with it. Calculate Conversion Rate based on this. For example, if you choose to use Clicks tracked in Google Analytics as the benchmark then measure Conversion Rate like this

Conversion Rate = Number of Affiliate Sales/Number of Clicks recorded in GA.

This may not be accurate but Conversion Rate is only useful as a baseline for future progress. As long as you stick with the same metric, you will compare like for like and get a good idea of how well your conversion optimisation efforts are going. After all, that's the whole point of this analysis.

2. You can't tie sales back to the sources of traffic that generated them (particularly keywords). A workaround is to customise event labels and custom affiliate parameter tracking like this:

traffic from widget queries -> widget pages -> clicks on aff links tagged with event label 'widget' AND custom affiliate tracking parameter 'widget' -> sale. 

That sale will be attributed back to link tagged with custom affiliate tracking parameter 'widget' which you can then confidently get the data for (clicks, source of traffic) from Google Analytics. Not ideal, not granular enough on a keyword level but better than nothing.