I have a Specialized Carve Comp 2013, and the front brake is very squishy. The brake system is Avid Elixir 1 hydraulic disc brakes. I wanted to inquire if I need to buy a specific kit to firm up the brake lever, or if there is a simpler, less expensive solution. Please advise, thank you.
This can be down to a misaligned caliper. Ideally when you apply the brake both brake pads should hit the rotor at the same time. If you can see a visible deflection in the rotor when the brake is applied then you need to adjust the caliper position. The loosen it off and apply the brake method rarely works for me, I get much better results lining everything up by eye. Also remember the caliper will try and move as you tighten the bolts up!
If you still have a squishy feel when the caliper is aligned then a bleed is the next thing to try.
You essentially need a brake service/bleed. Given the age of the bike it's possible that the brake pads and the hydraulic fluid haven't been changed for some years.
Brakes can feel squishy because of air in the hoses, but if the pads have worn down significantly, a big part of the squishiness/excess lever travel goes away when new pads are inserted. Due to the restored pad thickness, the fluid that has advanced down the hose is pushed back up towards the lever.
If needed, new pads can give a quick improvement to the lever feel, but it's also likely that the performance of the brake fluid has degraded in this time as well. So the brakes won't be back to full performance just yet. By performing or having a brake bleed done, the old fluid plus any air and/or moisture in the hoses is purged, leaving fresh fluid with its maximum boiling point restored and full performance once again.
A good mechanic may even clean the rotors of any previous contamination and finish with a bedding in procedure to marry up the new pads with the cleaned rotor, an important step for full performance.