I opened my thermometer to change the battery and the two red wires detached. Where do I reconnect them [solder them].

Solder it back on, use a low temp solder pencil.

Heating element connections r riveted together - u can't solder them as it would melt when running.
If the wire detached at the element it is near impossible to reattach it.
U would have to remove the rivet and replace it with the wires connected. My experience is that the mica that it is attached to is too fragile to get the ribvet removed.

on the m12 impact, when you look in battery well or after opening 3 wires for to battery connectors, check the whites soldering and how many wires are actually soldered in the whites line ... just fixed buddy's who had exact issue and only 3 strands were soldered, stripped about a centimeter more and soldered back on and boom issue solved.

remove four screws on back of unit,cut red wire going to round beeping speaker inside lower front faceplate. problem solved it will never beep again but still read temp correctly.

Change the batteries. If they are ok, often the micro-switch in the remote control has detached. It may need to be soldered in place again.

Have you tried replacing the disposable-lens filter ?
The manual is available at: http://brauntherms.com/downloads/IRT4020_4520_MNLA_U.0035_OM_25JAN11.pdf

The battery is a "G10-A CNB." To replace it, you need a flat surface, a good light, and a pair of tweezers or a small (i.e., jeweler's) screwdriver. A magnifying glass is also helpful to see how the battery is mounted.
1. Make sure the thermometer is turned off.
2. Find the side with the indentation and pry the upper half of the thermometer (the face) away from the body. The two are attached by a pair of wires. Be careful not to detach these during the remaining steps.
3. The battery is held in by a circular metal cover that is the same size and shape as the battery. The clip is soldered to the circuit board on opposite sides. Imagining that the solder points are at North and South of a compass, there will be two small flaps on the cover at the East and West positions. These are folded over the side of the battery and help hold it in place.
4. To remove the battery, push the battery near one of the flaps and force it out the opposite side of the clip where the other flap is. It may be necessary to pry up the opposite flap a bit before doing this.
5. Reverse the previous step to slide the replacement battery in under the clip.
6. Press the face of the thermometer back onto the body.
7. Turn the thermometer on to check whether the new battery is working.

first unhook the battery! then unbolt the cables from the back of the starter! be sure to mark them with masking tape, or something you can use to identify what wire goes where! then unbolt the starter bolts from the starter, and remove it! reverse this to put new starter back in!

Hi, disconnect the battery heads and reconnect. Yo can go to the fuse box in the bonnet and check the outer left margin. Detach each fuse one by one to check for burnt wires. You can replace this to make it work.
Cheers

Most likely you broke one or both wire from the backlight to the circuit board. Open the unit up and look for a black and red wire. Both should be soldered to the circuit board. LED's are used for the backlight and so it does make a different that the back and red wire are soldered to the circuit board correctly.