I recommend neutral cure silicone rubber adhesive, perhaps also using hotmelt glue to tack the LED in place while the silicone rubber sets. (An example of a commercial product where this dual adhesive approach is used is provided below).
Once set the silicone rubber bond will probably outlive the instrument.
By using neutral cure sealant you can be confident that the LED will not be damaged by sensible installation procedures. The common "acid cure" silicone rubber (smells like vinegar) runs the risk of damaging the equipment with acetic acid vapors.
"Hot melt" glue by itself will provide a 'quick fix' but will fail as a bonding agent in months to a year.
Cyanoacrylate will provide a good bond but you run the significant risk od the vapor from the liquid adhesive causing blemishes on the LED optical surfaces.
Details follow ...
A good compromise adhesive is neutral cure silicone rubber. It must be neutral cure if you care about the integrity of the equipment it is used on. This may be labelled as "RTV"* or "silicone sealing compound" etc. Be sure it is silicone rubber and not something somewhat similar. Reading the fine print will usually confirm this. If it doesn't say so anywhere use another brand, there are a number available. "Neutral cure" means that the compound does NOT release acetic acid as it cures. Most silicone rubbers do release acetic acid when curing - which gives them their characteristic "vinegar" smell. Acetic acid is significantly corrosive for electronic materials. There are two main types of neutral cure silicone rubber readily available. The most common type does bot bond well to polycarbonate but your meter case is unlikely to be polycarbonate and your LED will not be.
*RTV = "Room Temperature Vulcanising".
The acid cure silicone rubbers absorb atmospheric moisture and produce acetic acid during cure.
Where rapid setting is of great importance, use of eg hotmelt glue to lock a component in place, with silicone rubber providing the main long term bond can provide a superb result.
The photovoltaic panel in the product pictured below is retained in that manner. Hotmelt adhesive at the center of the panel is used to retain it when it is pressed into position. Neutral cure silicone rubber either side of the hotmelt and around the periphery is used to form the long term bond and seal. The panel is pressed home against guides and the hotmelt "blob" extrudes against the back of the panel, which is then held in place for the brief period required for the hotmelt to set. The neutral cure SR then can set "at its leisure". The end result is very satisfactory. Here the materials are ABS (body) and soldermasked FR4 PCB.
Hot glue or hotmelt has some excellent characteristics - most notably its very rapid setting time - but it is not a suitable long term adhesive mechanically. If you use it here you will probably be reusing it again within months to a year. Hotmelt sets extremelyrapidly (in the tiome it takes to cool thermally) but it fails to bond properly to most materials and what bond it does make becomes waker rapidly in weeks to months. You can tell when hotmelt has been used to hold mechanical parts n place in consumer equipment because after a while they rattle when you shake them - the glue debonds and the parts fall free unless also otherwsie constrained. One way that hotmelt works, which may be able to be made to work in your case, is making the piece of glue mechanically located regardless of its bonding ability. eg a "blob" of glue that extrudes through a hole and spreads out on the other side will lock in place as the larger piece cannot be pulled back through the hole. So if the glue was made of eg copper and could not be removed and locks the part in place adequately then hot melt may work for you.
Epoxy resin may work very well for you. You can get it in standard cure (1 hour+ tacking), 5 minute or "fast setting" (names used vary) versions - the latter tacking to a mechanically unworkable state in around 1 minute. If you get it on the LED lens you may mar the surface. It probably will still work if cleaned off promptly and well. It should be easy enough to use the fast setting type to make a bond which sets almost as rapidly as hotmelt glue does but with a much better bond life.
Cyanocrylate glues will bond the LED in most cases and probably bond to your case BUT run a significant chance of causing damage to the LED optical surface. CA glues emit a vapor when setting which can deposit on adjacent surfaces and cause damage. They can be excellent glues when used with understanding but are dangerous to use where the exact outcome cannot be controlled well. An aside: Formulations vary. The most effective CA glues which will bond to almost all plastics also can ignite when in contact with some organic materials. Ask me how I know :-).
There are a number of other adhesives which will work for you. But the characteristics of those above are liable to meet your needs without looking further.
Most LEDs in applications such as the meter will have an epoxy resin body. This is the well known "hard" clear plastic that most LEDs are made from. This material and other "hard" LED lenses are liable to be able to be glued by the materials mentioned above.
Some modern LEDs - usually high power ones - have silicone rubber lenses. These are softer - easily distorted or damaged with a fingernail or pen tip - and are much more sensitive to mechanical damage. They are also less suited for gluing with most adhesives. Silicone Rubbers will work. Others would need to be tested.
Dow Corning - example of silicone rubber materials for LED lenses
Dow Corning comparison sheet Epoxy versus silicones for LED lenses
Sekisui LED materials