RE: No power, burning smell
I have an inspiron 3847. The burning smell was coming from around where the processer is located and everywhere was fine.
As the power supply is next to processor/fan assembly and that there is no power, it sounds like the power supply has serious issues.
Try replacing the power supply unit to see if this makes a difference.
The Dell Inspiron 3847 Mini Tower can use generic ATX power supply units, with either a 24-pin or 20+4-pin main motherboard power connector, with or without the on/off switch.
You should be able to buy a compatible power supply from either a local or online computer store.
Note: You require a power supply with four SATA power connectors.
Best to buy a power supply with a depth of around 5.9".
If you have a Dell warranty in effect, contact Dell's Technical support, no warranty, then you will need to buy the replacement.