Having worked in insurance, I can give you a few pointers.
Firstly, state that you "may have to complain". Insurers hate complaints because they really complicate matters, are loads of work and must be tracked. I would advise not actually escalating it to a complaint until later as this may cause a delay as the actual process is quite convoluted. Mentioning the possibility of complaint sometimes makes people a bit more active. Try and resolve the issue, and if you aren't getting anywhere then make a complaint.
Maintain a friendly, assertive, polite demeanor. If you get angry and aggressive you'll not likely get far. Remember that the people on the end of the phone are both human and more knowledgeable about insurance than you. You want them on your side, not against you.
Make copious notes. If you can, record calls. If you are recording calls you will likely legally need to give them the option of not being recorded, so make sure you mention that you're recording each individual call as soon as you start speaking to the handler.
Refer to your notes and make sure you carry out actions you say you will. If you spend a few days sending something you said you'd email over that day, and you then chase them a few days after that they may not have had the document through their workflow yet. It also engenders urgency if you're acting promptly, and suggests that you don't really care if you're taking your time.
Get Names. This is an important step, as this gives the handler someone to talk to who may be familiar with your case. You may end up speaking to the same people more than once, so try and build a rapport if you do. "I like this guy" may lead to a bit more effort being put in, and a potentially better outcome.
In my experience, GoSkippy can be a bit slow to respond to things, so you'll likely have to chase them up. If you chase them up and say "I called on X date, discussing Y with Z and Z said they would do A, B and C. Has this been done yet?" it looks better than saying "I called about a week ago and spoke to one of your colleagues who said he'd do something for me. He's not done it, what's going on?",
As to a plan of action, I would split this into two points: Mis-sold policy and definition of commuting.
Mis-sold policy - If they truly gave your wife two options, then they messed up. The standard 3 offered by goskippy are Social Domestic and Pleasure (SDP), SDP+Commuting, and SDP+C and Business use. Other companies sometimes roll commuting into SDP as standard. On the comparison sites however, there are usually the three separate options, and if you used one to set the policy up and clicked "SDP Only" then you may be in trouble.
Social domestic and pleasure only DOES NOT include commuting. Whilst it is your responsibility to thoroughly check any documentation that comes through, it could be argued that if given two options between Business Use and SDP, then a reasonable person could be considered to assume that Goskippies definition of SDP did include commuting. Therefore, they need to prove to you that there were three options offered and that your wife specifically excluded commuting. IF they can't, then you should be able to argue that only two were offered and that commuting could have reasonably been assumed to be included. Use that term "reasonable person" btw, it's used in a lot of internal literature - at least at the insurer, I worked at.
not commuting - Firstly, clarify their definition of "commuting". If your wife was on her way to work afterwards, then they may well consider she was commuting. For instance, at present, I drive from my house to my son's childminders, then to my wife's work and finally to mine. My commute could be argued to be 1 minute (my workplace is probably a minutes drive from my wife's, but I can't park in her car-park), but if I were to have an accident between my house and my son's childminder (~15 mins drive) the insurers would probably consider that commuting.
If she was not on her way to work afterwards, and assuming your wife arranged her visit via text (or whatsapp, fb messenger or similar) you should easily be able to show that your wife had driven from a friends house to the childminder. If she was on a holiday day or was not working on that day, then that's also something you should be able to prove either with proof of her working pattern or proof of her holiday. If she doesn't have a job at all, then again, that's something that's provable.
Proof reigns king in claims, so if you can prove certain key facts then you should be on to a winner.