Fed up trying to get this sorted over the phone

Question

My parents are both pensioners and understand nothing about broadband. They joined BT ADSL broadband in the summer of 2011. I recently moved into the property and noticed the state of their broadband. It is terrible. I took it upon myself to try and sort it out with BT, thus begins a long and very frustrating adventure.
The problem is constant very slow speed, about 1mb, usually less in the evenings and constant disconnections throughout the day. Sometimes when the telephone rings. The master socket is ancient, we moved into the house in the early 1980's and the socket and wiring have never been touched by us. The socket has a hard-wired extension coming out of the side which runs upstairs to a phone in the bedroom. The extension cannot be unplugged. There is a filter on the main socket, which the router is plugged into and also a filter at the end of the extension in the bedroom.
This is the type of master socket we have
I phoned BT a few days before christmas, around the 22nd and explained the problem. They recognised there was a problem and said the would elevate it to 2nd line support. 2nd line support called a few days later and said they would get an engineer to call to arrange a time to come out. I never received a call from an engineer.
I left it until after christmas and called again on the 30th December. It appeared BT had no record of my previous call. I had to explain everything again and go through the same scripted process. I explained I had already been through all this and my case had been elevated to 2nd line support. He told me he had no record of that. Frustrated I just told him to give me a direct number for 2nd line support which he did.
I then phoned 2nd line support and wanted to know why I had not received a call from the engineer. They did not seem to know. I then had to explain the whole problem a 3rd time to 2nd line support. The chap said the problem was our ip profile and he would reset it. To wait 24-48 hours and he "promised" our speeds would increase and the connection would be stable. I waited for 48 hours and no change. I waited a further day and still no change. I checked on the BT speedtest and, according to that, my ip profile had not been changed at all.
I called 2nd line support yet again and explained the problem a 4th time. They told me my line was being capped and they would take the cap off and would call me back in 10 mins. In a few mins our broadband went down. When it came back I noticed our speed had jumped from 1mb to nearly 3mb (according to the BT website our estimated speed is 6mb). Hurray I thought, finally some progress. 2nd line support called back and said they would call back the next day (3rd Jan) to check how things were going.
The next day, our connection dropped again in the morning. I checked our speed again and it had dropped 600kbps to 2mb. I waited for the phone call from 2nd line support which never came.
I am sick of phoning and having to explain the whole problem every time I phone and being promised phone calls would be made that never materialise.
These are the speedtest results:
Before I called BT for the first time
Test from immediately after we were "un capped"
Test after connection dropped the next day
Current router stats
It is my belief that we need an engineer to come out and replace our master socket for a start but I am fed up trying to get through on the phone. I was hoping some member of staff who can actually get things done would be present on this forum.
Thanks for reading

Answer 1

Ok engineer came, two of them in fact, took one look at our master socket and swapped it for one of these
They tested the line and said they were getting speeds of 7.1mb on their equipment. Said it could take up to 3 days but we should see an improvement within 24 hours.
These are the stats after 4 hours of use on the new socket.
So fingers crossed in a few days we'll be up to 7mb or so. I'm assuming this speed should gradually increase of it's own accord? No profile needs resetting or anything?