Remember that recruiters are very busy with a lot of work and many candidates.
First of all, the last thing that you want to happen is to end up in the list of stalkers. You have already sent 2 e-mails in one week. This is enough follow-up for the moment.
Wait for a bit, without sending any e-mail, calling or anything and see if anything happens. In within 10-15 days you still receive no answer, then it is time to actively follow up again.
It sounds like to have a contact with a partner. This is like having the key to enter through the back door :-) I would rcommend to reach out to this partner (after 10-15 days of waiting, as mentioned above), saying that you have sumbitted your online test, you have already followed up with the HR in order to understand how long it will take to get the results, but since she hasn't answered after more than 2 weeky, you though to go directly to him, to see if he can give you perhaps more info.
I would avoid mentioning the technical problems story: in the consulting wold, we want people who bring solutions and not problems! Mention it only if it is absilutely necessary, for example, in case the HR comments that you took very long to complete the test or similar.
I hope this was helpful, good luck!
