Sweet peas are cool weather annuals, which germinate when the ground temperature is between 55-65 degrees F. Sweet peas need about 50 days of cool temperatures to bloom. To increase germination, Tamu.edu suggests nicking the outside coating of a sweet pea seed to allow rapid hydration of seeds. Use a nail clipper to score the sweet pea seed coat. Germination should occur in about 10 to 21 days. Once germinated, the plants do best with a temperature of 70 to 80 degrees.

When to Plant
It looks like the fall/winter is best. The Orlando Sentinel suggests planting in November so they'll start vining in January and begin blooming in March. But Reneesgarden.com says if you miss planting in the fall, you can plant in mid-January or February. ehow suggests sowing seeds directly in the ground from late fall right into early spring.

Indoor Planting Tips

1. Start seeds in jumbo packs or 4-inch pots.
2. Fill pots with a soil free seed-starting mix.
3. Plant 1 inch deep and two to 3 inches apart using two or three seeds per pot.
4. Put the pots in a cool, dark place for until germinated. Once germinated bring into the light.
   
5. Sweet peas grown in containers will need regular fertilization every few weeks.
6. Transplant outside as soon as they have two or three pairs of leaves.

Outside Planting Tips

1. Choose seeds that say on the label it flowers during the cooler weather or under short days (per Orlando Sentinel).
2. Plant near a trellis (or another support system) so they can climb and produce long flower stems.
3. Improve sandy soils with organic matter, like rich compost or well-rotted manure.
4. Plant seeds one inch deep and 4- to 6-inches apart.
   
5. Keep the soil moist until the seeds germinate then water when the surface is dry to the touch.
6. Add a 2- to 3-inch mulch layer to conserve moisture.
   
7. Lightly use garden fertilizer monthly or fertilize at least twice by drenching the soil with a mixture of a tablespoon each of fish emulsion and liquid kelp per gallon of water.
8. Control aphids with a soap spray or blasts of water.
9. Select an area with full sun to part shade but make sure to provide afternoon shade in very hot areas of Florida.
10. Good air circulation and avoiding watering in the late afternoon and evening
    will help prevent mildew.
11. When seedlings are several inches tall, thin seedlings to about 5- 6 inches apart.
12. Once they bloom pick often to encourage new growth.
    

Hope this helps!
For more info check out the sources below:
You might also want to ask at http://centralfloridagarden.blogspot.com/

Sweet peas like cooler weather. They barely grow well in Indiana because of the hot, humid, unpredictable summers; they are definitely something to try in Florida in the winter! I'd plant them late fall there; keeping in mind it's going to take a couple of months for them to get big enough to bloom--you don't want it to get too hot before then, so planting earlier is the right thing to do.

I would suggest hydroponics. Here are a few links to check out.
I would have written more on this but I am suddenly pressed for time. It's a solid option for you.

http://hydrogarden-hydroponic-gardening.blogspot.com/2008/05/hydroponic-gardening-peas.html
http://www.healthgoods.com/shopping/gardening_products/Aerogrow_Aerogarden_Snow_Peas_Seed_Kit.asp <-- the aerogarden might be a good option for you.
http://www.articlesbase.com/videos/5min/40002954
http://www.scribd.com/doc/3857987/Hydroponics