As a monk, after failing to hit someone with a dart throw, can I make an unarmed strike against an enemy close to me using my bonus action?
In order to use the Martial Arts ability (that lets you make an extra unarmed strike as a bonus action) you need to:
When you use the Attack action with an unarmed strike or a monk weapon on your turn you can make one unarmed strike as a bonus action.
And a Monk weapon is:
shortswords and any simple melee weapons that don't have the two-handed or heavy property.
The Dart, being a ranged weapon, does not qualify as a Monk weapon.
But sometimes yes
Once you hit level 5, the Extra Attack feature gives you the option to attack twice. You only need to attack with a Monk weapon once to qualify for the bonus attack from Martial Arts, so if you throw a dart and then punch someone (or strike with a different Monk weapon) you'll be able to use the bonus attack.
Alternatively, the Flurry of Blows ability (which gives you 2 bonus attacks for a Ki point and is available at level 2) merely requires you to take the Attack action, so in that case it doesn't matter what weapon you use, you'll get the extra attacks.
Also, some Monk paths (such as Way of the Kensei) and optional class features (such as Dedicated Weapon) let you pick additional Monk weapons. If you select the Dart as a Monk weapon, you'll also be able to use Martial Arts after making an attack with it. There's also the Ki-Fueled Attack optional feature, which lets Martial Arts trigger whenever you spend Ki, which you could combine with Focused Attack to do it.