Google limestone cleaning. There are many things you should not use on limestone.

I know you are interested in cleaning the grout, however, you need to know what to not use on the limestone.

I have no clue what Finish Power Ball dishwasher detergent would do to limestone, it does clean grout good by soaking/scrubbing with stiff grout brush. If very dirty, will need a repeat or another repeat.

I have cleaned several rooms of grout/porcelain tile with good results using the above. I do need to repeat the process again and again if necessary to get clean enough to put a sealer on the grout. All depends on how dirty the grout is.

Most hand grout brushes require too much muscle power to use, buy an "Unger" grout brush, the grey one from a janitor supply place. Put brush on handle of desired length to scrub. The brush is about 8" long and beveled, fits in the grout line just right to get maximum scrubbing. Think the Bristles are nylon and are very, very, very stiff but do not scratch the tile. For floors I put a long broom handle on the brush making it easier to scrub standing up.

We have a lot of limestone on the first level of our apartment. Love the look. Sorta hate the maintenance. Given your wise ban on scrubbing bubbles or Tilex, it seems you already understand the stone's unforgiving nature (can be the most porous and susceptible to stains).

When was the last time it was sealed? In a wet area, it's going to require a watchful eye -- with glasses on. It sounds to me like you may be beyond grout restoration and need a grout replacement. That poses its own risks depending on the initial installation (I've learned) but you're right to be prioritizing the tile versus the grout.

Personally, I'd stay away from anything at the supermarket and be wary of formulas for "all natural stone". I 'm gaga over Mira Clean (Miracle Sealants Company). They have really good info and specifics for each type of stone. A friend recommended a place called Marble and Granite Care Products. I've not tried it but they also have a grout-specific kit.