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Since it's so easy to access being nearly empty of water, I'd first try and wipe and brush off the algae manually first so that there's less for chlorine and your filter to deal with. You can then decontaminate with a high chlorine level (see
Using Chlorine In a Spa. Because the spa is attached to your pool, I don't think you can use a product like Spa System Flush since you don't want that water to get into the pool, but for just superchlorinating that's not a problem. If you can completely isolate your spa and its circulation from the pool, then Spa System Flush would be OK and would clean out your pipes of oils and/or biofilms, but you'll need to drain the water out of the spa to waste, not into your pool and not have it spillover during that process either.
If you think your spa isn't in terrible shape except for the algae, then you could just shock with chlorine (say to 20 ppm) instead of the high superchlorinate (100 ppm) or use of Spa System Flush. I don't know your spa's history so don't know what shape it's in. Basically, if you do just a shock and then find balancing the spa that the 24-hour chlorine loss from the spa isolated from the pool is more than 25% FC drop (at hot spa temperatures), then you've got a chlorine demand problem (if the pool is exposed to sunlight, then look at the overnight loss which should be no more than a 15% FC drop, again at hot spa temps).