It is hard to say what is wrong, but if there is no spring rewind action, the rewind spring could be broken.
Try this: With the starter housing lying upside down in your lap, turn the wheel inside it to your left, about 6 times. If it turns to the right with spring pressure, the spring is probably not broken. If it turns with some resistance, and then suddenly loses the pressure, the spring is probably broken, or un-hooked.
the procedure for fixing it is too long to describe tonight, but the best way, if you are not a mechanic type, is to have a mower repair shop fix it.

See if this will help. Don't take the spring out unless you want a nightmare, unless, of course, it's broken. I doubt it.

Usually the easiest way to replace a starter rope is to remove the top metal shroud from the mower, to which the rewind assembly is riveted. It's probably held on by 4 bolts and possibly the oil tube is attached to it. Disconnect the oil tube by removing the one screw that attaches it to the shroud and let it in place in the engine to avoid a mess. On the common Briggs engine, the gas tank has to be removed first. Remove the majority of the fuel using any approved safe method. Remove the 4 screws on top and 1 screw on the underside of the tank of a Briggs engine.

On the Tecumseh engine, the tank is often affixed to the shroud by light pressure. This example is a Tecumseh shroud and the rewind assembly is held in place with two external screws, shown in the highlighted area. Removing these screws will allow the rewind assy. to be lifted off, without removal of the entire shroud. There is also an aluminum crimp on the rope.

I always apply Never Sieze to any bolts removed, before reassembly.

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Remove the old rope by pulling the remainder all the way out until you see the knot at the end of it, which is still remaining in the assembly. If the rewind spring was getting weak, you may be able to turn the pulley one more revolution tighter for more recoil strength. Measure the length of rope needed and replace with the same diameter rope.

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Line up the slots and put a screwdriver through the closest slots to hold the recoil spring at that position, so it doesn't rewind. Now push the end of the rope backwards and cut the knot and remove the rope completely.

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Push the new rope through the same holes you pulled the old rope through and tie a single knot, trim off the extra rope on the outboard side of the knot, put a few drops of fingernail polish on the knot and let it soak into the fibers. Install the pull handle using the same single knot and nail polish.

Now you can hold the rope, remove the screwdriver, and slowly feed the new rope into the rewind pulley until the slack is gone.

Re-attach the engine shroud and oil tube, pull the rope out of the housing and secure the pull handle at the handlebar. If you have used the same diameter rope as the original, you will be set to go. If the rope is a larger diameter, you may have binding problems, a short pull, and will have to remove some rope at the pull handle end. If your rope is a thinner diameter, you'll be OK, but it won't last as long.