The power will go into the female barrel jack on the arduino
OK. It looks like you haven't yet done anything that might damage your Arduino. But, I see some reasons for concern.
1) The voltage across the LEDs when they are on will only be 5V. It looks like you intend to connect the anodes of the LEDs to timer output pins on the Arduino, and to connect the cathodes to ground. The timer output pins are only at 5V relative to ground when they are turned on. You'll probably see the LEDs operate, and dim - and it looks like you have seen that. But, you won't get full brightness from the LEDs. If you run the assembly at 9V, it'll be dimmer still. But, if you're happy with what you see, that may be just fine.
If I'm wrong - if you intend to connect the LEDs to the 12V source - stop and reconsider. Arduino inputs can't stand up to voltages more than a little bit above V
CC. They'll work as long as they're turned on, but when you turn them off, unpleasant things will happen to the input pins on the Arduino.
2) The specs on your LED assemblies are 20 mA at 12V. With an LED forward voltage drop of something between 2.0 and 3.3V, depending on color, among other things, the resistor calculates to about 430 to 500 ohms. With 5V rather than 12V, the LED current will be around 3.4 to 7 mA. I can't tell how many LEDs you have on a single pin, but, with a total of 16 LEDs on 6 pins, some pins must have at least three. There's some possibility that you will exceed the 20 mA limit for current through an Arduino output pin, again, depending on the color. I think that red LEDs are likely to draw more current, but the actual conditions will depend on how the manufacturer implemented his assemblies, and he doesn't seem to say. That 20 mA limit is a little bit soft, in that the absolute maximum, stress-rating for the pins is 40 mA. But, to run 24/7, long term, I'd very much recommend staying below 20 mA. You can find the ratings for pin currents in the datasheet under "Electrical Characteristics." It seems unlikely that you'll exceed the limits described in the notes to the table, "DC Characteristics," but it's worth a look.
3) Hooked up like this, all of the output power used by the device - the power that runs the Arduino, and the power that illuminates the LEDs - goes through the Arduino's 5V regulator. We can't tell exactly how much power that is, because of uncertainty about how the LED assemblies are made. You could relieve some of the stress on that regulator, and operate your LED assemblies at their nominal voltage, by connecting the LEDs like this: Anode to 12V; cathode to the open collector output of a ULN2803, or something like it; ground pin of the ULN2803 to ground; and the base of the ULN's darlington pairs to the Arduino output pins. The output pins will have to deliver only a little bit of current to the ULN, and the power for the LEDs will come directly from the 12V supply, bypassing the Arduino's regulator. Note that you may or may not need resistors between the Arduino pins and the ULN, depending on which one you use - some have resistors inside the IC, and some don't, and you need to have resistors.
If a ULN-style interface IC isn't readily available, you can use several transistors instead, with resistors, of course, between the transistor bases and the Arduino pins.
I will be soldering on the rest of the LEDs on site when we install the project next week. I let the whole thing run for about two days now, seems to be doing fine although the chip gets a little warm.
It'll get warmer, maybe downright hot, when you add the rest of the LEDs. I think you'll be happier, and your project will operate more reliably, if you use a ULN-style buffer or discrete transistors to switch your LEDs. The Arduino draws something like 30 to 40 mA; multiplied by the 7 volts the regulator has to drop, that would amount to about a quarter watt - not bad.
[Edit: add this] With 16 LEDs at 7 mA - my calculation for the worst case with the LEDs between an Arduino pin and ground - that goes to about a watt. It'll get hotter.
If you decide to go with an interface IC or with transistor switches, and have trouble with wiring or calculations, post. We'll walk you through it. If you decide to stay on your present course, there's a fair chance that it will work just fine if you don't connect more than three LEDs to any pin, and you don't mind low brightness. You'll have to weight the consequences of device failure against the extra effort of doing something different.