I worked it a sterile antibiotic facility for six years, so I'll try to share some basic facts about sanitizers.
Soaps help break the surface tension on your skin. Washing with soap and water will remove dirt, skin oils, and germs. Soap residues also act a dessicant - drying out some of the remaining vegetative cells. Although warm water (say 110 F) may result in cleaner hands, the higher temperature doesn't kill many germs. It's just not hot enough for long enough. (Note: some strains of E coli have a fairly limited temperature range, and are killed at lower temperatures. But remember that if they are thriving at 98.6, then 110 F isn't too far out of their comfortable range).
Anti-microbial soaps are a joke. Washing works by removing large concentrations of germs, not by killing the microbes. These soaps can cause germs to build resistance to the trace chemicals in the formula, potentially leading to more robust (and dangerous) strains of microbes.
Commercial / retail hand sanitizers typically work by oxidation. Witness the popularity of the gelled alcohol sanitizers. Alcohols, hydrogen peroxide, acetic acid (vinegar), chlorine compounds, phenols, and quats work by oxidizing the cell wall of the microbes. Literally, the microbial cell walls are ripped apart when exposed to these substances. Efficacy of the various products is variable, depends on the product, chemical concentration, and exposure time.
Household bleach is a VERY effective sanitizer, even at very low concentrations - such as what you used. There are two things to note. First, chlorine is a toxic substance. At higher concentrations it is toxic to almost all life forms. Second, chlorine is an important trace element for both plants and animals. You ingest it every meal, in the form of salt (NaCl). So chlorine isn't bad in and of itself, but strong concentrations can pose health / environmental issues. Some people object to its use simply because of the environmental problems caused by its manufacture and distribution.
Alcohol based sanitizers are only effective against vegetative cells, while most of the other sanitizers are effective against both vegetative cells and spores (a tough, "seed" form of bacteria and mold). Chlorine kills some spores, depending on the particular microbe resistance, solution concentration and exposure time.
On the other hand, alcohol works almost instantaneously. It kills the majority of vegetative cells in the time that it takes to evaporate from your hands. Other compounds require a longer wetting period.
Iodine is an effective sanitizer. It is commonly used in beer and milk processing plants, to sanitize piping loops. I don't know all of the reasons that it is not used as a hand sanitizer, but I suspect that the smell and color turn people off. It really stains skin!
Finally, remember that water is the universal solvent. Sanitizers work best in the presence of water. For example, a 90% alcohol solution will not be as effective as a 60% alcohol solution.
To keep all of this in perspective, the current focus on aseptic environments is much overblown. Polishing all surfaces in your house with Chlorox wipes isn't going to make you live longer, or have a healthier home. The air you breathe is full of microbes, hitching a ride on the thousands of dust particles in front of your face. Soil is a witches brew for microbes. You couldn't survive without the organisms in your intestines, processing the food you just ate.
Scientists have conducted a number of studies that prove that children have healthier immune systems when growing up in a "less clean" environment.
You CANNOT effectively remove or kill all microbes from your hands. The surface of your skin is extremely rough. Compared to the size of various microbes, the ridges and valleys of your skin look like mountains! Microbes actually live and reproduce between the layers of dead skin cells (some are known as facultative anaerobes - they don't need oxygen, but can survive in its presence).
Skins cells constantly flake away from your body - THOUSANDS a minute. Everytime you move an arm or leg, you generate a literal stream of dead skin cells.
If you ever get a chance, look at your fingertips under a stereo microscope (20X - 40X would be good). Look at your hands after getting them dirty, then look again after washing them. It will change your perspective, guaranteed!
One last note. The real concern is not dirty hands - even after going to the toilet. Those bugs just came out of your body, and in the right context, are a healthy part of life.
The concern is transferring certain microbes into a growth medium (certain foods at certain temperatures) that would allow it to proliferate. After multiplying exponentially, some strains doubling the number of cells every 20 minutes, the food becomes a serious danger.
So wash your hands, with water, water and soap, a mild chlorine solution, or a readily available gelled alcohol. All of them will work. The goal is to REDUCE, not eliminate the population. If you feel that the washing was not especially effective, then just be careful about your food preparation. With common sense and awareness, all will be well!