My comments for what they are worth. Note: I don't expect you to agree with me, and certainly may not!
Reading your first few sentences, I wonder why you even have a nanny. It is your parental choice to have someone help with childcare of course, but if you find the arrangement to not be working well, from how you describe things you could change to an alternative type of provider quite easily.
I do not know what is written in the contract between you and your nanny. Templates from agencies are often quite lacking, and need to be used as a starting point for a contract. Section on Duties will usually need to be amended/added as well as what constitutes Gross Misconduct. As your nanny has been with you for a few months, perhaps it's time to have a Contract Review and let your employee know your expectations - plus document those expectations. Also good opportunity for employee to give you feedback as well, are they happy in their job for instance?
Outings - Winter is a hard time for anyone with young children. Getting out and about can be tricky, but effort should be made so that children (and carer) get daily exercise.
Perhaps you can sit down with your nanny and make up a Outings Folder, which contains leaflets of local places to visit, maps for local walks, swimming pool timetable, transport timetables etc. If I get stuck for something to do, my little ones love to sit on a train. Simple things like going on a train, bus, boat, hovercraft are often greatly enjoyed by children.
I feel you were very brave to let your nanny bring her Dog to work. If that isn't working out, then be firm and tell her that her dog is no longer welcome. It's your house after all.
Sickness - It's Tuesday, and she's called in Sick for the entire week? I hope she's told you what illness she has. When she returns, do a Return-to-Work interview, so that you can record why she was off sick. As a caring employer you want to know about any thing that may be causing difficulties, though try not to pry too much if she does not want to reveal personal medical information to you. You may find that by discussing her time off, you will establish how she feels about the job... as she sounds unmotivated, perhaps she isn't enjoying the job.
In my view, it is unreasonable to expect your nanny to have higher standards than yourself. Saying that though, some nannies may well have higher standards. It all depends on the person - who you choose to employ.
You are the employer, so if you do not want your sons watching DVD at all during the day, you can say so. However I would try to avoid a blanket ban, instead start more softly by restricting the time period.
I got through today without the children watching any DVD, though they did protest a bit. Instead we all went on starfall.com, and leant about the alphabet, phonics, and the 3 year old learnt how to use the Printer, plus practice mouse skills. Computer Time can be similar to DVD Time in certain ways, but does tend to be more educational, and can involve everyone. Point I am trying to make is that there are alternatives to DVD/TV.
With regard to food, if you expect your nanny to make things from scratch, then during those times the children will not get much attention and thus may be dumped in front of the DVD, to help avoid health and safety issues arising. Some multi-tasking is possible depending on the circumstances, though young children and kitchens do not tend to mix well. However, if you are able to keep an eye on the children, then certainly no reason why your nanny could not make up fresh purees and prepare lunch/tea.
And how many sickies do you think is reasonable?
Is there ever a reasonable number?
BA Cabin crew were about to strike over that sort of issue this week, weren't they?
The key I feel to sickness is to monitor it closely. Do return-to-work interviews following sickness, so that you discuss with your employee regarding their time off. Pay for sickness should be clearly put in the contract, and as the employer you certainly do not have to pay full salary - SSP can kick in quite quickly.