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Interstingly, I hook L1 to E1 directly and L2 to E2 directly and still no heat.
I doubt this.
Most people will rarely be able to notice 'HEAT' as in 'WARM' or 'HOT' from a 5500 watt element in a hot tub. This is the equivalent of trying to heat two to four hundred gallons of water with 4 blow dryers.
Which means, it takes time. You would need to do this for at least 30 minutes and monitor temperature rise over that period using an accurate digital thermometer. If you had an ammeter and could measure the current draw of the system, I'm sure that once you powered it up configured this way, you would've seen at least 20+ amps being drawn from your line, which means you're definitely heating.
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This must be an element problem then, right? It is brand new! Before I order another, is there a way to make sure everything else is working?
You've tested your element already and found you had continuity... 12 ohms which is normal.
Your element is ok.
Since you noted:
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when heater indicator light is on and there are no error messages
I would suspect that the T-92 high limit relay (Item C in your photo) is defective.
This is a common high failure item and if you follow the line from the heater terminal to it, then measure the voltage on the other side of the contact. Better yet, since you've replaced the element, go ahead and replace that relay while you're at it - it should've been replaced when you replaced the element anyway.
The coil voltage for it should be in the 115v range otherwise it's a special order item:
http://spapartsnet.com/Electrical-Parts ... 1_0_1.html
You say you don't know anything about electricity, well you've done pretty good so far explaining your situation which is a lot more than I can say for the tons of calls we get daily.... just check back and post what you find so that the folks that follow the forum can chime in with you and check to see if what they're saying has helped.
Last thing... (Has nothing to do with your troubleshooting issue):
Whatever fool wired this hot tub without a ground? (Hope it wasn't you my man - in commercial electrican circles jobs like this make the bulletin boards all the time for humorous anti-diy fodder)
If I got a call on this one, I'd either refuse to work on it, work on it and leave with the entire electrical supply disconnected, fix the ground issue (and install a GFCI), or call the freakin cops

If I were you I wouldn't do anything else until a proper ground line is run from the house box all the way to the tub - NO, do not substitute a ground rod - it's not the same and leads to a false sense of security.
This thing is not safe.