Hi and Welcome.
I am also working part-time, and grantd, it’s not easy, but it is possiblle. Finding a routine that works for your family is the key. If your children have been to PS as mine were, you may find it difficult to alter their thinking towards a Monday to Friday week with summers off, but using your Saturdays and summers is definately one way of making it work.
I work mornings, Monday to Friday until noon. During this time my boys are expected to do their independent work and chores. I get home about 12:20, we have lunch and try to get to our afternoon work by 1pm. Most of this time is family work — bible. history, geography, etc (about 1hour to 1 1/2 a day). Then we check morning work and take care of any corrections, or guided learning that needs to be done. We do our family read aloud in the evening. Sometimes we only have one literature book going, sometimes we have more. Sometimes we’ll have a family one as well as a seperate one for each of the boys at bedtime — thankfully we just have the two.)
In our province we do not have to keep records, and there is no specified number of days or hours we have to log. If we did it could be a little more tricky.
If I were you I would focus on getting your 15 and 13 year old going, getting them as independent as possible, before worrying too much about working with your five year old. You can accomplish a great deal with a child that age in a very short amount of time. Read to your little ones as much as you can (your olders can share in this), work for 5 or 10 mins a day each on printing and math and reading as he/she shows readiness/interest, and include him/her in famly studies where appropriate. Don’t try to do too much at once. Get outside as much as you can.
You may have to start small — start with the basics, get a routine going. Figure out what the daily minimum you can live with will be, and start there. Be realistic in what you can expect from yourself as well as your kids.