Just give yourself a goal, 15 minutes of ANY exercise per day.
I don't agree that you have to start out a weight loss journey with an epic/intense work out. Just commit yourself to an exercise goal and do it. You can of course do more than 15 minutes a day if you want, but the main thing is you do 15 minutes no matter what. That's your lowest limit and you should be happy with yourself when you achieve it. Anyone can fit 15 minutes into their day. If you can give yourself 15 minutes to eat food, you can give yourself 15 minutes to exercise.
I personally do that, and I take 1 rest day a week. I also watch my calories and make sure I'm always in deficit to how much I actually need to maintain. At my weight and my height I need to eat around 1800 calories for maintenance. So I make sure I never go over 1800 no matter what, that's my upper limit. And I always make sure I'm in deficit every day, so that at least I'm losing something.
You also need to mentally remind yourself that weight loss is going to take a long time. I'm not expecting myself to lose 2-3lb a week. I'm happy with even a 0.5lb weight loss a week or even maintenance. I think it's important to remind yourself that it's a long journey and that even if you fall off the wagon the one day, you get back on the next day. If you fail the next day, you try again the day after.
And of course... stay on 3FC because the help and support here is awesome.
Good luck!
P.S. I'm so jealous you have an elliptical at home! That was my favourite machine when I used to gym a few years back. I always got awesome results on that. For a start, you can just do elliptical for 15 minutes a day and watch your calories so that you're in deficit. If you get bored of that routine, just switch it up with the stationary bike, or even do some sprint walking around the block.