Q:

How to secure the drill bit to the tool body?

I have two pieces set of precision hand drill. One of them had the drill bit became loose and detached, which makes it no longer usable.

Product information (copy of text in English found on the packaging, which may be subject to translation errors by the manufacturer):

2 pcs. set: 1.0 mm, 1.5 mm (approx. 0.04/0.06 in.)
Usage: For drilling of handicraft, plastic model, hand works
Materials: High speed steel, Iron  

The tool body has a hole with very small clearance, which makes it almost impossible to put any adhesive inside to secure the drill bit back in place (the adhesive glue will either resist the drill bit to be inserted again in place, or will cause most of them to ooze out). I also tried using some epoxy glue at outside surface that was sandpapered, but failed to withstand the hand-drilling force.
Is there anyway to secure the drill bit to the tool body; alternatively, if repairing as such is not possible, is there other way to secure the drill bit to something else for hand use (not machine)?
P.S.: The main question is "how to secure (repair)". Although, I am open to other lifehack solutions that may be possibly better than repairing this tool.
P.P.S.: The alternative method should restore the functionality of the precision hand drill, preferably at very low cost (a lifehack is required).

Update (13 Aug): Thanks for all the suggestions; although, the repair seemed to be more difficult than expected with each new answer. Meanwhile, I have figured out a low-cost lifehack and I think I can self-answer my own question soon.
Update (15 Aug): By using the same epoxy and the glue applying method suggested by the first answer, the drill bit apparently remain secured. But I am not confident with the adhesive; I happened to found another "free" drill bit of same size (1.5 mm) from my junk parts' box, so I worked on the alternative method.

A:

You say you tried placing adhesive in the hole but it is too tight a fit: there is nowhere for the excess adhesive to go.
Instead, apply the adhesive to the shank of the drill bit. Then any surplus adhesive will be pushed further up the shank and not impede entry.
Thoroughly degrease the drill bit and the hole first, with repeated application of solvent.
Looking at a zoom of the image, it seems as if the shank of the drill bit is hexagonal, and the hole too. If it weren't for that, I would say it won't be possible to make a satisfactory repair.

If you can't repair it, you can re-use the drill bit if you have a pin vice (vise) with an adjustable collet or chuck, such as this.

An adjustable pin vice (vise) is a good investment anyway, because drill bits go blunt and break (especially small ones), so all you need to buy is another bit, not the whole tool.

A:

These tools are almost never made well these days, as you have found; drill bits pushed into a metal handle of mediocre grade alloy and relying on friction alone to grip. The drill bit's surface is smooth and grip is poor
I've generally had a rule with tools that I buy cheap ones and replace those that fail with ones of better quality. Additionally, the handle of this tool is not much fatter than the bit that it holds, so it doesn't offer appreciable extra torque over just using the bit in your fingers. Look for a replacement tool that has a handle with a slim part (for faster rotation) and a fatter oart( if you need greater torque). If you break a good quality version of this tool, leverage the manufacturer warranty to replace it and consider that you're probably drilling something that requires use of an actual chucked drill
It would however be possible to repair this tool thus:

Use a grinder to grind a flat spot into the side of the drill bit shaft, at the end that is inserted into the tool handle. The size of the flat spot depends on the torque you anticipate applying; bigger flat spot allows more torque

Apply epoxy to the flat spot. Because there is now space for the epoxy, sufficient quantity will remain on the drill bit when is is inserted into the handle. The epoxy's job is not to resist the turning force but simply to retain the bit in the handle
Insert the drill bit to the handle, put the handle in a vise and gently crush the handle into the flat spot. Leave the epoxy to set

If you're concerned about misalignment of the drill bit (I wouldn't be) consider alternatively drilling a small hole in the side of the handle so that the hole aligns with the center of the flat spot. Load the epoxy into the handle down the large hole where the drill bit goes - it will succeed now that the air can escape out the small hole. Push the bit into the hole so epoxy squeezes out the small hole. Clean up and wait for set. The epoxy will form a key between the small hole and the flat spot preventing slip, but it won't be able to resist as much torque as the crush method

Apologies for the crudeness of the images; they were drawn with a fat finger on a cellphone that has a thick glass screen protector and it was hard to be precise, but hopefully they convey the meaning 

A:

Braze it or Silver Solder it
If glues and epoxies fail, for a more permanent solution, you can try brazing the drill into the holder. You would need a torch setup to do it. If you don’t have the equipment or care to buy it all, maybe a jeweler, welding shop, or friend who has a torch could braze it for you(but probably for more than what a new one would cost). 

