I would recommend Red Mulberry (Morus rubra) they have very good sweet fruit and are very juicy. Do Not get White mulberry they can be invasive and are very weedy, and there fruit is kind of bland and less sweet, they also carry a root disease that can damage or kill the native Red mulberry trees.

Mulberry is dioecious Producing male and female flowers on separate plants, so both is needed for pollination.

But they can be monoecious, with male and female flowers on different branches of the same plants.

There is no way to tell between them when they are young because they have no flowers, they start to make flowers at about 10 years of age but some have been known to as young as 4 years old. Only way to know for sure is if a cutting off a mature tree is rooted or grafted then the nursery could tell you if it's male, female, or monoecious maybe, but I'm not sure if any do that. But if they grow mulberries from seed they won't know until about 10 years later.

I would recomend the Russian mulberry. It has the purple juicy berries. The native red ones have very sparce fruit. There is a cultivar called Illinois everbearing that is available from nurseries.

I believe 'Illinois Everbearing' is supposedly a sterile hybrid(so no unwanted seedlings), is monecious(self fruitful), and produces an abundance of fruit over a very long time in summer and starts producing at a young age, so it might be your best bet. You can get it from several places, but I know that Stark Brother's has it and so does forestfarm.

Stark Bro's order page for 'Illinois Everbearing'

forestfarm page for 'Illinois Everbearing'

I would absolutely stay away from any cultivar of the Asian White Mulberry(Morus alba) which is an extremely invasive weed which will spread for at least a mile all around where you plant it. I'd also recommend the native Red Mulberry(M. rubra), since it's doesn't seem to be anywhere near as weedy as the Asian White Mulberry and is a much nicer tree IMHO.

I'm a mulberry affectionado, and my recommendation is based on growing and fruiting a dozen or so named varieties.
"Illinois Everbearing" is the 'gold standard' for those of us who can't successfully grow M.nigra - and that will include you - it is a heavy producer of big, tasty berries with a good sweet/tart balance and good flavor, and it bears over a long period of time - 6-8 weeks in my orchard, from early June to late July. Grafted trees will begin bearing in a very short time - just a couple of years, and well worth the expense compared to 'rolling the dice' with a seedling tree that may turn out to be a non-fruiting male or one with small, less-than-desirable or tasteless fruits.

I've never tasted a bad M.rubra - the berries are tasty, typically a bit larger and plumper than those of IE, but you get one 'wave' of fruiting, and that's it. Productivity is a fraction of what you'd get with Illinois Everbearing.

Stay away from M.alba, including 'Russian' mulberry - it's a weedy, invasive species, with fruit quality inferior to M.rubra or the M.albaXrubra hybrids, like Illinois Everbearing.

"Stearns" is probably my next favorite variety, followed by "Colllier". "Wellington" bore well this spring, but it's the first decent crop in the 12 years I've been growing it, so I'm not yet ready to sing its praises.

I have a 5 yr old russian mulberry tree that produced fruit the first year after planting subsequently it produces profuse flowers in early spring but no fruit , any thoughts as to how I can induce it to produce fruit

If nothing else has changed--like a male tree in the neighborhood that was fertilizing yours being cut down--you might ease off feeding the tree. You should first check your soil for fertility, but I have heard that mulberry trees don't fruit well if the soil is too nutrient or fertilizer rich.

I hope that this helps.