Thank you for attaching the images. Unfortunately, they don’t reveal sufficient detail for identification.

Flies and gnats are numerous every spring but it seems especially so this year. All belong to the nuisance category in that they don’t cause disease or damage plants. You didn’t mention how you used DE; it’s effective only if it damages the insect’s exoskeleton (outer surface) allowing it to dry out. Thus, it’s typically effective against insects that walk through it but not if the insect is flying.

Among the strategies against flies, “Sticky fly paper or ribbons are effective at eliminating a few flies in relatively confined areas but are not effective enough to manage heavy infestations or to provide control in an outdoor setting. Inverted cone traps containing fly food attractants can be readily purchased commercially and are effective when they are not competing with nearby garbage or animal wastes.”  See “Flies” http://ipm.ucanr.edu/PMG/PESTNOTES/pn7457.html. (Notice that this page includes several flies which are not at your place: canyon flies, face flies, and stable flies.)

You didn’t mention if you lived in a residential or rural area. If rural, excellent possible breeding places for flies include manure and wet straw. This information about house flies has numerous management suggestions about half way down the page: http://entomology.ifas.ufl.edu/creatures/urban/flies/house_fly.htm

Another option is to invite several pest control companies for onsite evaluations and management suggestions. Guidelines for “Selecting a Pest Control Company” are at http://npic.orst.edu/pest/selectpco.html.