Really silly question - but the wi-fi on this Inspiron is enabled ? Surprising how often the obvious gets overlooked.
Go into 'Device Manager' and look for any yellow exclamation marks against your wi-fi adapter - click 'Start - RIGHT click on 'Computer' - click 'Properties' - click Device manager in the top left hand corner'. You will see an item 'Network adapters' with a + beside it. Click on the '+' to expand it and you should see your wi-fi adapter listed. If it has any sort of mark against it, double click on it and see if you can find any info on the problem. In any case, make a note of the maker and model. Problems in this area are usually caused by the drivers. Uninstall the drivers for it, and on another computer, go to the Dell web-site and download the wi-fi drivers for your Inspiron, copy them to some convenient location on the faulty computer then install them. Re-boot, and with any luck your wi-fi will be working again.
It is possible that the wi-fi unit itself has failed. If it is not physically built into the motherboard it may be possible to replace it with a new one. But the simplest and cheapest solution I have found is to use an external USB wi-fi dongle. These are widely available and fairly cheap, starting at under £UK 20 and going up.