Nails that peel are lacking in sufficient levels of oil and moisture that trasnmits from the nail bed and constantly 'bathes' the nail plate. Always wear gloves when using household cleaning solutions or when having your hands in water. Water and cleansers will wick the natural oils and moisture from the skin and nails,leaving them dry, peeling and prone to splitting and breakage.
Ask yur local nail salon if they retail Solar Oil. If so, purchase a bottle and apply it to the cuticles and nails at least twice a day, every day. You should notice an improvement in less than a month. Instead of wearing polish, buff your nails to a high shine with a 3-way buffer. This will help to seal the nail plate layers while they are 'healing'.
If you cannot find Solar Oil, try looking for it at an Ulta outlet or purchase Botanical Oil from your nearest Sally Beauty Supply.
For more information on your natural nails, check out [url="http://www.hooked-on-nails.com"]http://www.hooked-on-nails.com[/url] (an approved web link.)