Complaint sub-issue: Credit card company isn't resolving a dispute about a purchase on your statement

On XX/XX/XXXX XXXX I signed up, online, for a {$29.00} monthly fee for a basic study course for my granddaughter at XXXX. Since she would be the one using the account I put it in her name ( XXXX XXXX ) and her email. The credit card information of course was in my name along with my all my information. 
Within a week I noticed on my credit card statement I was being charged {$190.00}. Logging into the website I found I was being charge {$190.00} a month instead of the {$29.00} monthly. ( Attachment : study account info from website ) Their only form of communications I could find was through their online portal. I sent a request for help XX/XX/XXXX in which I stated to contact me at my phone number and/or my email. Giving them a few days I did not hear from the XXXX so I filed a dispute with my credit card XX/XX/XXXX. 
Elan credited {$170.00} back to my account. They sent a letter to me requesting proof that I had signed up for {$29.00} monthly. Since this was an online transaction I did not have that proof. By that time though I had contacted my granddaughter and found that XXXX was emailing her instead of me. I sent that email to Elan showing where I complained to the company, XXXX stating I had signed up for {$29.00} course. Along with that I answered the other information that I could provide. 
I received another letter on XX/XX/XXXX from Elan stating they still needed proof and requested I response to XXXXXXXX  response. They included attachments from XXXX which showed their policies and that the account had been canceled and a refund was sent. It shows on XXXX attachment that the refund included the {$170.00} chargeback. It also states a partial refund of {$29.00} which was reimbursed to my account XX/XX/XXXX. I responded to ELANs letter stating the matter was resolved with a refund and attached email stating the money had been refunded by XXXX. I felt the matter had been resolved. 
Then on XX/XX/XXXX I received a letter from Elan, stating they could not help with my dispute and they charged my account {$170.00}. Immediately I reached out to XXXX  in regards to this informing them the chargeback money had been resent to them on XX/XX/XXXX and needed to be refunded. XXXX finally worked with me directly stated in an email the money was refunded and I needed to work with the bank. 
I placed a call to Elan requesting this be looked into. I did not note my first call but was told the person working the dispute would call me back within several days. I called on 4 other occasions, XXXX, XX/XX/XXXX, XXXX, XXXX in regards to this usually being told of a returned phone call. I finally received a call back after requesting the manager call but unfortunately missed the call. When I attempted to return the call I was told quite rudely Elan can not help me because I have no proof of what I purchased or as Elan stated in a letter " there is insufficient information supplied to support your claim of incorrect amount '' Not knowing what to do I filed a complaint with FTC and the XXXX XXXX XXXX XXXX. I received communications back in regard to the XXXX  complaint where again I was told Elan looked into it and found they had done nothing wrong. 
At this point I am confused. First off, I was under the impression that credit card companies were responsible for helping with incorrect charges. Also if XXXX refunded the {$190.00} on XX/XX/XXXX why am I am still out {$170.00}. And most important, why is ELAN or XXXX not helping but instead pointing fingers at each other. 

I have spent well over {$170.00} worth of my time on this matter but to me there is a huge principle at stake here. From the feelings I have gone through most people would have walked away and some company, Elan or XXXX, would have additional money at hand.