Hey there. Nice car. As much as I don't want you leaving Detail City for your detailing advice,
Welcome to Team Integra! - Team Integra is a site that DetailCity.com sponsors and they have some awesome detailing advice very specific to Integras. I've been a member over there for several years ever since I bought my '99 GSR sedan. I have since sold it, but I still like to check in on them to make sure they are still keeping them clean.
Something else that might help is the engine detailing section of
this article on my site. I actually owned the Integra at the time I wrote it and is very specific to the Integra. That might help some.
However, with all that being said, I might as well also try to answer your question here.

The important thing to remember about engine detailing is that you aren't putting out a fire. Don't just spray water all over the place and assume that is going to get the job done. You need to take your time and probably get your hands dirty. For really neglected engine bays it may take more time than one that is just dirty.
What I would do to start is use a foaming degreaser product that you can buy at any auto part store. Spray that on the painted surfaces around the perimeter of the engine bay. Avoid spraying things like exposed filters, the alternator, and electrical connections.
Then use low pressure with a garden hose and rinse it down. Put your thumb over the end to give it a little pressure if necessary, but remember that you are likely going to have to go in with a brush to really get it clean.
With a bucket of soapy water, and probably a bottle of Simple Green handy, you would then hand scrub everything that is still dirty. I use toothbrushes, wheel brushes, wash mitts, and anything else that will help me get into the little nooks and crannys. My philosophy is that if I can see it, then I can clean it. I take special pride in my engine detailing.
A good detail on a neglected engine could take an hour or two, depending on the condition. I spent two hours on an old Monte Carlo engine simply because there was oil and grease sprayed everywhere. It took some time, but it looked great when it was done. Be patient and take your time with it.