Rates are �8 - �10 for an half hour walk. Add time to pickup and dropoff and it will take you a whole hour for one job. Don't be fooled into combining dogs on one walk or you can be in serious trouble with expensive dogs. Although you can charge more for taking 2 or 3 dogs from the same place or combine those that have been properly socialised. Some people won't pay you full rate if you combine jobs though.
To clear �600 per month you need to earn �60 per day to cover tax. insurance etc. If you are able to work 8 hours per day, 5 days per week then yes, it can be done. I don't know what your health reason is but If however you can walk for 40 hours per week does that negate your reason for being unable to continue your 'exiting' previous vocation??