There are a variety of metrics that you can provide.
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Increased referral traffic. If your links are good and are coming from pages and sites that people read, then your referral traffic should always rise.
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Increased traffic to specific landing page. If you are targeting the links towards specific landing pages, then their traffic should go up, right?
3) Increase in rank for targeted keywords. This is easier if the client is long term. We always tell clients its going to take at least 2 weeks for rankings to increase. This is my least favorite metric, but it does help clients understand.
Another thing that I have found is that clients need to know as much about where you failed as where you succeeded. We always provide a list of pending or failed link attempts. This shows them that the work we do is difficult, helps them understand our efforts.
It sounds to me like what could have been improved was the initial planning period. If you had helped set expectations from the start, writing down your plans and KPI's you might not have this issue.
That being said, sometimes no explanation is good enough. If you feel like you've done everything you can and the client is still not happy with your answers, it may be best for you both to go in a different direction. Nobody wants to give up revenue, but sometimes it is for the best.