Infinity 2 up and down speed, very slow, huge ping...

Question

Hello wonderful friendly people, i'm looking for a bit of help to narrow down the cause of the terrible internet connection I have.
I live in a flat in a new build block in Bristol, just moved in with the owner who has had a few different connections with different providers in the last few years but have always been really bad service. He is now paying for Infinity 2 but we experience massively varying internet speed, as low as 0.25 Mb and as high as 20Mb download, but upload seems to be a bit more stable.
Streaming netflix, downloading files and general browsing is somtimes OK, but mostly terribly slow. It seems the speed goes up and down constantly, hitting rock bottom very often. Playing games is an absolute nightmare. Ping as low as 5, then shoots up to 500 meaning we can't play at all.
In short, paying a lot for terrible internet. The connection is through the phone line, we have a Homehub 3 but with an ADSL modem, i'm not sure we should be paying for Infinity if it's going through ADSL though? I thought Infinity was only phone line for the last bridge between house and local cabinet?
I think the problem is with dodgy cabling in the flats. From my experience this is where the problem points to but I want to be sure before I go to BT and ask for them to fix their line problems.
Here are some results from the last few days from speedtest.net
Can anyone help me with tests I can do to find where the problems lie, i.e. ping tests, trace routes and the like?
Cheers!

Answer 1

Download speedachieved during the test was - 3.89 Mbps
 For your connection, the acceptable range of speedsis 16 Mbps-77.43 Mbps .
 Additional Information:
 IP Profile for your line is - 77.43 Mbps
Upload speed achieved during the test was - 6.77Mbps
 Additional Information:
 Upstream Rate IP profile on your line is - 20 Mbps
Thanks